computational complexity of some optimization problems...

Linköping Studies in Science and Technology

Dissertation No. 1854

Computational Complexity of some Optimization Problemsin Planning

Meysam Aghighi

Linköping UniversityDepartment of Computer and Information Science

SE-581 85 Linköping, Sweden

Linköping 2017

c©Meysam Aghighi, 2017ISBN 978-91-7685-519-5ISSN 0345-7524URL http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-136280/

Published articles have been reprinted with permission from the respectivecopyright holder.Typeset using LATEX

Printed by LiU-Tryck, Linköping 2017

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Computational Complexity of some Optimization Problemsin Planning

ABSTRACT

Automated planning is known to be computationally hard in the general case. Propositionalplanning is PSPACE-complete and first-order planning is undecidable. One method for ana-lyzing the computational complexity of planning is to study restricted subsets of planning in-stances, with the aim of differentiating instances with varying complexity. We use this method-ology for studying the computational complexity of planning.

Finding new tractable (i.e. polynomial-time solvable) problems has been a particularly impor-tant goal for researchers in the area. The reason behind this is not only to differentiate betweeneasy and hard planning instances, but also to use polynomial-time solvable instances in orderto construct better heuristic functions and improve planners.

We identify a new class of tractable cost-optimal planning instances by restricting the causalgraph. We also study the computational complexity of oversubscription planning (such asthe net-benefit problem) under various restrictions and reveal strong connections with classicalplanning. Inspired by this, we present a method for compiling oversubscription planning prob-lems into the ordinary plan existence problem. We further study the parameterized complexityof cost-optimal and net-benefit planning under the same restrictions and show that the choiceof numeric domain for the action costs has a great impact on the parameterized complexity.

We finally consider the parameterized complexity of certain problems related to partial-orderplanning. In some applications, less restricted plans than total-order plans are needed. There-fore, a partial-order plan is being used instead. When dealing with partial-order plans, oneimportant question is how to achieve optimal partial order plans, i.e. having the highest de-gree of freedom according to some notion of flexibility. We study several optimization problemsfor partial-order plans, such as finding a minimum deordering or reordering, and finding theminimum parallel execution length.

The research presented in this thesis has been partially funded by the National Graduate School of Com-puter Science in Sweden (CUGS).

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Populärvetenskapligsammanfattning

Huvudtemat i denna avhandling är studiet av beräkningskomplexitet förautomatisk planering. På en hög nivå kan man betrakta planeringsproblemetså här; man utgår från en värld eller ett system, som ett obemannat fordoneller en autonom robot, och antar att det finns vissa handlingar som ändrartillståndet i systemet. Man antar dessutom att det finns ett initial- och ettmåltillstånd. Planering är problemet att hitta en sekvens av handlingarsom tranformerar initialtillståndet till måltillståndet. En sådan sekvens avhandlingar kallas en plan. En planerare är ett datorprogram som löserplaneringsproblem, och planerare baseras ofta på sökning genom tillståndeni systemet. Vanligtvis vill man optimera sökningen och hitta en plan som äroptimal med avseende på något kriterium. Vad man vill optimera beror påtillämpningen men det är ofta att hitta en kortaste eller billigaste plan.

En viktig fråga när man studerar automatisk planering är hur snabbtman kan hitta lösningar. Ett sätt att analysera denna fråga är att studeraberäkningskomplexiteten för olika planeringsproblem. Komplexitetsteoriär den gren av datavetenskapen som försöker klassificera hur svåra olikaberäkningsproblem är, det vill säga analysera hur mycket resurser (såsomtid och minne) som behövs. Man likställer ofta effektivt lösbara problemmed de som kan lösas i polynomisk tid: ett problem kan lösas i polynomisktid om det finns en algoritm vars tidsåtgång begränsas av ett polynomi indatas storlek. Det grundläggande planeringsproblemet är exempelviskänt att vara PSPACE-fullständigt. PSPACE är den komplexitetsklass sominnehåller alla de beräkningsproblem som kan lösas genom att användapolynomiskt mycket minne och de PSPACE-fullständiga problemen utgör desvåraste problemen i PSPACE. Man tror att det inte finns några PSPACE-fullständiga problem som kan lösas i polynomisk tid—detta är ett viktigtolöst problem inom teoretisk datalogi—och man förväntar sig därför inte attdet generella planeringsproblemet kan lösas i polynomisk tid.

En metod för att analysera beräkningskomplexitet av planering äratt studera begränsade delmängder av planeringsinstanser i syfte attsärskilja fall med varierande komplexitet. Vi använder denna metodgenomgående i denna avhandling. Så kallad “oversubscription planning”är en typ av planering där man mäter hur nära ett givet måltillstånd

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man kan komma. Vi studerar beräkningskomplexiteten för denna typav planering under olika restriktioner och avslöjar starka kopplingarmed vanlig planering. Kostnadsoptimal planering är planering där varjehandling har en kostnad och målet är att hitta en plan med minimal kostnad.Dessa handlingskostnader kan hämtas från olika numeriska domäner såsomde positiva heltalen eller de rationella talen. Vi studerar effekter avvalet av numeriska domäner och noterar intressanta komplexitetsmässigaskillnader. För kostnadsoptimal planering identifierar vi dessutom en nyklass av instanser som kan lösas i polynomisk tid. Slutligen studerar vikomplexiteten för olika problem inom så kallad partialordningsplaneringdär man tillåter planer som inte nödvändigtvis är sekvenser av handlingar.

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Acknowledgments

First and foremost, I want to thank my advisors Peter Jonsson and ChristerBäckström. It has been an honor to work with and learn from them for thepast three years. They have taught me how well the academic research isdone. I am specially grateful to Peter for his great help and Christer for hisvaluable comments in preparing this thesis.

The members of TCSLAB and IDA contributed immensely to my per-sonal and professional time as a PhD student at LiU. I specifically thank Si-mon Ståhlberg, who was both a colleague and a friend, Amir Aminifar, Bi-man Roy, Arian Maghazeh, Adrian Horga, Rouhollah Mahfouzi and VictorLagerkvist.

I am also grateful to the administration at IDA that facilitated my workin less direct ways. I specially thank Karin Baardsen, Anne Moe, Marie Jo-hansson, Inger Norén and the director of graduate studies, Professor PetruEles.

Lastly, I would like to thank my family for all their love and encourage-ment. For my parents who raised me with a love of science and supportedme in all my pursuits. For my loving, supporting, encouraging and patientwife Zeinab whose support during my whole PhD is so appreciated. Thankyou.

MEYSAM AGHIGHI

Linköping UniversityMay 2017

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List of Papers

The thesis is based on the following papers:

1. Meysam Aghighi and Peter Jonsson.Oversubscription planning: Complexity and compilability.In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelli-gence, Québec City, Québec, Canada, July 27 -31, pages 2221-2227, 2014.

2. Meysam Aghighi, Peter Jonsson and Simon Ståhlberg.Tractable cost-optimal planning over restricted polytree causal graphs.In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelli-gence, Austin, Texas, USA, January 25-30, pages 3225-3231, 2015.

3. Meysam Aghighi and Christer Bäckström.Cost-optimal and net-benefit planning - A parameterised complexityview.In Proceedings of the Twenty-Fourth International Joint Conference on Arti-ficial Intelligence (IJCAI), Buenos Aires, Argentina, July 25-31, pages 1487-1493, 2015.

4. Meysam Aghighi and Christer Bäckström.A multi-parameter complexity analysis of cost-optimal and net-benefitplanning.In Proceedings of the Twenty-Sixth International Conference on AutomatedPlanning and Scheduling (ICAPS), London, UK, June 12-17, pages 2-10,2016.

5. Meysam Aghighi and Christer Bäckström.Plan reordering and parallel execution – a parameterized complexityview.In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence,San Francisco, California, USA, February 4-9, pages 3540-3546, 2017.

In addition to the above papers, the author has also contributed to the fol-lowing publications:

1. Meysam Aghighi, Christer Bäckström, Peter Jonsson and SimonStåhlberg.Analysing Approximability and Heuristics in Planning Using the

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Exponential-time Hypothesis.In Proceedings of the Twenty-second European Conference on Artificial Intel-ligence (ECAI) , the Hague, Holland, August 29th - September 2nd, pages184-192, 2016.

2. Meysam Aghighi, Christer Bäckström, Peter Jonsson and SimonStåhlberg.Refining Complexity Analyses in Planning by Exploiting theExponential-time Hypothesis.In Annals of Mathematics and Artificial Intelligence (AMAI), pages 157-175,2016.

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Contents

Abstract iii

Populärvetenskaplig Sammanfattning v

Acknowledgments vii

List of Papers ix

Contents xi

I Introduction 1

1 Thesis Overview 31 About this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Brief Summary of Papers . . . . . . . . . . . . . . . . . . . . . . 4

2 Computational Complexity Theory 71 Computational Complexity . . . . . . . . . . . . . . . . . . . . 72 Parameterized Complexity . . . . . . . . . . . . . . . . . . . . . 11

3 Automated Planning 151 What is Planning? . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Planning Model and Formalism . . . . . . . . . . . . . . . . . . 183 Planning Problems . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Background and Contributions 231 Paper 1: Oversubscription Planning . . . . . . . . . . . . . . . 232 Paper 2: Causal Graphs for Tractability . . . . . . . . . . . . . 243 Papers 3, 4: The Effect of Cost Domain in Cost-optimal and

Net-benefit Planning . . . . . . . . . . . . . . . . . . . . . . . . 254 Paper 5: Plan Optimization . . . . . . . . . . . . . . . . . . . . 26

References 29

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II Papers 37

Paper 1 39Oversubscription Planning: Complexity and Compilability . . . . 391 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Planning Framework . . . . . . . . . . . . . . . . . . . . . . . . 413 The Partial Satisfaction Problem . . . . . . . . . . . . . . . . . . 424 The Net Benefit Problem . . . . . . . . . . . . . . . . . . . . . . 445 Summary of Complexity Results . . . . . . . . . . . . . . . . . 466 Compiling NBP into PE . . . . . . . . . . . . . . . . . . . . . . . 497 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Paper 2 55Tractable Cost-optimal Planning over Restricted Polytree Causal

Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 Defoliation of Polytrees . . . . . . . . . . . . . . . . . . . . . . . 615 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 Correctness and Time Complexity . . . . . . . . . . . . . . . . 657 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Paper 3 71Cost-Optimal and Net-Benefit Planning – A Parameterised Com-

plexity View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 Parameterised Complexity . . . . . . . . . . . . . . . . . . . . . 733 Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 Complexity of Cost-optimal Planning . . . . . . . . . . . . . . 765 Complexity of Net-benefit Planning . . . . . . . . . . . . . . . 796 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Paper 4 87A Multi-parameter Complexity Analysis of Cost-optimal and

Net-benefit Planning . . . . . . . . . . . . . . . . . . . . . . . . 871 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882 Parameterised Complexity . . . . . . . . . . . . . . . . . . . . . 893 SAS+ Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 Parameterised Cost-optimal Planning . . . . . . . . . . . . . . 915 Complexity Results for COP . . . . . . . . . . . . . . . . . . . . 926 Complexity Results for Net-benefit Planning . . . . . . . . . . 987 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Paper 5 109Plan Reordering and Parallel Execution – A Parameterized Com-

plexity View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Parameterized Complexity . . . . . . . . . . . . . . . . . . . . . 1113 Planning Framework . . . . . . . . . . . . . . . . . . . . . . . . 1124 Least-constrained Plans . . . . . . . . . . . . . . . . . . . . . . 1135 Parallel Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166 Reordering Parallel Plans . . . . . . . . . . . . . . . . . . . . . 1187 Parallel k-Processor Plans . . . . . . . . . . . . . . . . . . . . . 1198 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

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Part I

Introduction

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1 Thesis Overview

1 About this Thesis

The main topic of this thesis is the study of the computational complexity ofautomated planning problems. Automated planning has been studied by theartificial intelligence community over the past fifty years. The intersection ofcomplexity theory and planning has been studied since the early work byChapman [21] which proved that first-order planning is undecidable. Sincethen, the complexity of planning has attracted a lot of interest.

This thesis is a compilation of five papers. It consists of two parts. Thefirst part contains four chapters. This chapter gives an overview of the thesis.Chapters 2 and 3 give an introduction to computational complexity theoryand automated planning, respectively. In Chapter 4, the background of eachpaper and its contributions are summarized. The second part consists offive peer-reviewed conference papers published between the years 2014 and2017.

In order to read and understand the contents of the thesis, a basic knowl-edge in discrete mathematics and graph theory is expected from the reader.Also, familiarity with planning and computational complexity theory is veryhelpful, in order to understand the contributions of each paper, the motiva-tion of the discussed problems and the intuition behind the proofs. Somegood references for complexity theory are Arora and Barak [4] and Gareyand Johnson [31], for parameterized complexity Flum and Grohe [30] andDowney and Fellows [27], and for planning Ghallab, Nau, and Traverso [33]and Geffner and Bonet [32].

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1. THESIS OVERVIEW

2 Brief Summary of Papers

This section gives a brief summary of the five papers. More informationabout the background and contributions of each paper can be found in Chap-ter 4.

Paper 1

This paper provides a complexity analysis of the oversubscription planningproblems, many sub-classes are introduced under the PUBS and Bylanderrestrictions and a detailed complexity map is presented for them. Moreover,a polynomial-time compilation of net-benefit planning problem to plan exis-tence is presented using a novel counter-based reduction.

Paper 2

This paper presents a new class of tractable cost-optimal planning instances.Our algorithms uses a novel concept of variable isomorphism and showsthat every planning instance with: (1) bounded domain size, and (2) polytreecausal graph with bounded diameter, is polynomial-time solvable.

Paper 3

This paper gives a parameterized complexity analysis of the cost-optimaland net-benefit planning problems. The two problems are studied with dif-ferent cost domains for action costs and under Bylander and PUBS restric-tions. Our results show that cost-optimal planning for positive integers (Z+)is not harder than length-optimal planning while it becomes harder if we usenon-negative integers (Z0) or positive rationals (Q+).

Paper 4

This paper continues paper 3, but this time using a multi-parameter com-plexity analysis. The paper identifies the parameters that affect complexitythe most, such as plan length, maximum denominator of action costs andsum of goal utilities.

Paper 5

This paper continues the work of Bäckström [5] on reordering partial-orderplans by applying parameterized complexity theory and introducing param-eters like the height and width of a partial-order. The paper analyzes planoptimization using two criteria: order size and parallel length of a plan. Italso introduces a new problem: finding the minimum parallel length of a

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2. Brief Summary of Papers

partial-order plan when there is a bounded number of processors. We showthat this problem is significantly harder than the unbounded case.

The papers appear exactly as in the conference proceedings. However,there has been some minor formatting adjustments. The papers of this thesiscan be divided into categories from several points of view. From the goal per-spective, the first four papers consider cost-optimal and net-benefit planningwhile paper 5 is on plan optimization. From the complexity analysis point ofview, papers 1 and 2 use traditional complexity while papers 3, 4 and 5 useparameterized complexity. From the planning domain perspective, papers 1,3 and 4 use syntactic restrictions on classical planning, paper 2 uses both syn-tactic and semantic restrictions and paper 5 goes beyond classical planningby considering partial-order plans.

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2 ComputationalComplexity Theory

1 Computational Complexity

Computational Problems

Computational complexity theory is a branch of computer science that classi-fies computational problems according to their inherent difficulty. There areseveral types of computational problems, such as decision, counting, search,optimization, etc. An optimization problem asks for best solution amongthe set of all possible solutions and a decision problem asks for a yes or noanswer, i.e. is there a solution for a given problem instance? Formally, a prob-lem instance and its solution are considered to be strings of bits (i.e. membersof the set t0, 1u˚). Simple encodings can represent general mathematical ob-jects such as integers, graphs, vectors, etc. A computational problem is aset of problem instances with a solution for each one of them, i.e. a functionf : t0, 1u˚ Ñ t0, 1u˚. Note that this solution can be an arbitrary mathe-matical object, like a list of numbers or a set of shortest paths in a graph.A decision problem is a special case of computational problem where f is aboolean function f : t0, 1u˚ Ñ t0, 1u which is often identified with the setL f = tx : f (x) = 1u that is called a language. In complexity theory we aremostly interested in decision problems. We give two concrete examples.

Example I: Satisfiability

Let Φ be a formula in conjunctive normal form (CNF), e.g.

Φ(x1, x2, x3) = (x1 _ x2)^ ( x1 _ x2 _ x3)

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2. COMPUTATIONAL COMPLEXITY THEORY

Φ is satisfiable if there exists a truth assignment to the variables x1, x2 andx3 such that the value of the formula Φ(x1, x2, x3) is true. For example, x1 ÞÑ

TRUE, x2 ÞÑ FALSE and x3 ÞÑ TRUE is a satisfiable assignment for the aboveformula.

The decision version of SATISFIABILITY (SAT) problem is defined as fol-lows:

INPUT: A CNF formula Φ with variables in X = tx1, x2, . . . , xnu.QUESTION: Is there a satisfying truth assignment for Φ?

A truth assignment can be viewed as a function α : X Ñ tTRUE , FALSEu andthe decision problem as a function f : t0, 1u˚ Ñ t0, 1u. f takes a string inthe input that represents a CNF formula and the set X. Then it returns 1 ifthe formula is satisfiable, otherwise it returns 0. L f is the set of all strings onwhich f returns 1, i.e. the set of all satisfiable CNF formulas.

Next, we present an example based on an optimization problem:

Example II: Traveling Salesperson Problem

The optimization version of TRAVELING SALESPERSON PROBLEM (TSP) isdefined as follows:

INPUT: A graph G = (V, E) and a function c : E Ñ N.QUESTION: Find the minimum value k P N for which aHamiltonian cycle of cost k exists, i.e. there is a permutation(v1, v2, . . . , vn) of vertices in V such that (vn, v1) P E, for every1 ď i ď n´ 1, (vi, vi+1) P E and(

n´1ÿ

i=1

c(vi, vi+1)

)+ c(vn, v1) ď k

The optimization problem can be viewed as a function f : t0, 1u˚ Ñt0, 1u˚. f takes a string in the input that represents a TSP instance, whichis a graph and a cost function for the edges. Then it returns a string in theoutput that represents the minimum integer k for which a Hamiltonian cycleof cost k exists, or a special designated character that means no Hamiltoniancycles exist in the graph.

Model of Computation

In order to measure the complexity of computational problems and analyzethe resources required for algorithms and computers, we need to define amodel for computation. There are several models for computation such asthe random-access machine, the finite-state machine, the Turing machine, thelogic circuit, etc. The Turing machine was proposed by Alan Turing in 1936and is the most commonly used model in computation. This is because of theChurch-Turing thesis; a hypothesis stating that every "reasonable" model of

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1. Computational Complexity

computation can be simulated by a Turing machine with only a polynomialslowdown in run-time in terms of the input size. Please note that this wasnot the original Church-Turing thesis, but a result of the extended Church-Turing hypothesis [4]. It is also believed that for any computer algorithm, aTuring machine can be constructed to simulate the algorithm’s logic [70].

A Turing machine consists of an infinite memory tape divided into cells,a set of symbols and a finite table of instructions. The machine’s head ispositioned over a cell and reads the symbol there. Then, based on the readsymbol and the current position of the machine in the table of instructions,the head writes a symbol in the cell and then moves either one cell to theleft or one cell to the right. Then, it either proceeds to the next instruction orhalts the computation.

Formally, a deterministic (one-tape) Turing machine M is described by a7-tuple (Q, Γ, b, Σ, δ, q0, F) where [45]

• Q is a finite, non-empty set of states that M can be in, including a des-ignated initial state (q0) and a set of final or accepting states (F Ď Q).

• Γ is a finite, non-empty set of symbols that M’s tape can contain, includ-ing a designated blank symbol b (the only symbol allowed to occur onthe tape initially often at any step during the computation). Γ is calledthe alphabet of M.

• Σ Ď Γztbu is the set of input symbols.

• δ : (QzF)ˆ Γ Ñ Qˆ Γˆ tL, Ru is a function describing the instructionthat M uses in every step. L and R denote the left and right movementby the machine’s head. δ is called the transition function and denotesthe table of instructions. If δ is not defined in the current state and thecurrent tape symbol, then the machine halts.

A Turing machine accepts the initial tape contents, if it eventually halts in astate from F.

There are several variations of the Turing machine. The two most com-mon are deterministic Turing machine (DTM) and non-deterministic Turingmachine (NTM). In a non-deterministic Turing machine, the instructions’ ta-ble allows more than one action in each situation and the transition func-tion is replaced by a transition relation. For instance δ may contain both((q, a), (q1, b, L)) and ((q, a), (q2, c, R)). This is to be interpreted as: if the ma-chine is in state q and its head reads the symbol a, it has the choice of goingto state q1, write symbol b and move left, or go to state q2, write symbol c andmove right. An NTM accepts an input if there exists some choice of actionsthat leads to the designated halting state. These variations are in principleequally powerful but when the resources (such as time and space) are re-stricted this equality breaks down.

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Complexity Measures

The two most common complexity measures are time and space. The run-ning time of an algorithm, intuitively, is the number of basic operations per-formed by it, where a basic operation takes a fixed amount of time to per-form. The run-time of algorithm A on input x is formally defined as thenumber of state transitions of the corresponding deterministic Turing ma-chine for the algorithm, on input x. The time complexity of an algorithm isthe run-time as a function of the input size (number of bits needed to repre-sent the input). Similar to time, the space complexity is the amount of work-ing storage an algorithm needs. The complexity is often reported asymptot-ically, as the input size goes to infinity. For instance, if an algorithm takes3n5 + 4n2 log n´ 2n units of time for an input of size n, we say that the timecomplexity of the algorithm is O(n5), using the big O notation. This is be-cause for large values of n, the value other terms are insignificant comparedto the term 3n5. Since the time complexity of an algorithm may vary withdifferent inputs of the same size, we usually consider the complexity as themaximum value over all inputs of size n. This is referred to as the worst-casecomplexity analysis.

Reduction and NP-completeness

Reduction is a way to show that one problem is at least as hard as anotherproblem. Intuitively, problem A is reducible to problem B, if an algorithmthat solves problem B, can be modified to solve problem A. There are sev-eral types of reductions and the most common form is the polynomial-timemany-one (Karp) reduction which is usually referred to as the polynomial-time reduction.

Formally, we say that a language A Ď t0, 1u˚ (equivalently, a decisionproblem) is polynomial-time reducible to a language B Ď t0, 1u˚ denoted byA ďp B if there exists a polynomial-time computable function f : t0, 1u˚ Ñt0, 1u˚ such that for every x P t0, 1u˚, x P A if and only if f (x) P B.

A complexity class is a set of computational problems that can be com-puted within a given resource. For instance, DTIME( f (n)) is the class of alldecision problems that can be solved by some DTM running in time O( f (n))where n is the input size and f is some function. The two most well-knowncomplexity classes are P, the class of problems solvable in polynomial timeby a DTM, and NP, the class of problems solvable in polynomial time by anNTM. Informally, a decision problem belongs to the class of NP if and onlyif its instances for which the answer is “yes” have efficiently (polynomial-time) verifiable proofs. The two definitions are equivalent because the firstpart in both contains a guess about the solution, which is generated in a non-deterministic way, while the second part contains a deterministic algorithmthat verifies or rejects the guess as a valid solution to the problem.

There are many other complexity classes such as EXPTIME, the class ofproblems solvable in exponential time by a DTM, PSPACE, the class of prob-

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2. Parameterized Complexity

lems solvable in polynomial space by a DTM, NL, the class of problems solv-able in logarithmic space by an NTM, #P, the class of counting problemsassociated with decision problems in NP, etc. The complexity classes areusually distinguished by: (1) The type of the computational problem (de-cision, counting, etc.), (2) The model of computation (DTM, NTM, Booleancircuits, etc.), and (3) The resource constraints (polynomial time, logarithmicspace, etc.).

Let C be a complexity class. Problem A is C-hard if there is a polynomial-time reduction from every member of C to A. This means if there is an ef-ficient algorithm for A, then this algorithm can also solve all problems in Cwith only a polynomial blowup in time. Moreover, A is said to be C-completeif it is C-hard and additionally in C. One may say that the C-complete prob-lems are, intuitively, the hardest problems in C. Both SAT and TSP decisionproblems from Examples I and II are proved to be NP-complete [31, LO1,GT37]. This results in the optimization of TSP to be NP-hard, because a so-lution to the optimization version of TSP will immediately solve its decisionversion.

The P vs NP problem (P ?= NP) is a major open problem in computer sci-

ence with a million dollar prize by the Clay Mathematics Institute. It asks ifevery problem whose solution can be verified in polynomial time by a com-puter can also be solved in polynomial time by a computer. Most researchersin the area believe that the answer to this question is negative (i.e. P ‰ NP)and this would imply that many common problems cannot be solved effi-ciently. However, a positive answer would have enormous practical conse-quences; for instance, many cryptography algorithms rely on the difficultyof certain problems [60].

2 Parameterized Complexity

In traditional complexity analysis, the time complexity is measured only interms of the input size. However, many problem instances consist of two ormore parameters that contribute differently to the complexity of the problem.

Example: Vertex Cover vs Dominating Set

The problems VERTEX COVER and DOMINATING SET are defined as follows:

VERTEX COVER

INPUT: A graph G = (V, E) and a positive integer k.QUESTION: Does G have a vertex cover of size at most k, i.e. isthere a set of vertices V1 Ď V such that |V1| ď k and for everyedge (u, v) P E, either u P V1 or v P V1?

DOMINATING SET

INPUT: A graph G = (V, E) and a positive integer k.QUESTION: Does G have a dominating set of size at most k, i.e.

11


is there a set of vertices V1 Ď V such that |V1| ď k and for everyvertex u P VzV1, (u, v) P E for some v P V1?

Although both problems are NP-complete, the input parameter k con-tributes differently to the complexity of the two problems. Let n be thenumber of vertices. The best known algorithm for VERTEX COVER solvesthe problem in time O(1.2738k + kn) [22], while the best known algorithmfor DOMINATING SET is the brute force algorithm of trying all k-subsetswhich takes O(nk+1) time. This is a significant difference; the ratio of nk+1 to1.2738k + kn is more than 1016 for n = 150 and k = 10. ˝

The observation of such differences between NP-complete problems mo-tivated a new and more fine-grained form of complexity analysis: param-eterized complexity analysis. Parameterized complexity tries to find howdifferent parameters of a problem contribute to its hardness. For instance,parameter k in VERTEX COVER usually has a much smaller value comparedto n in practice. Therefore, even an exponential algorithm that runs in timeO(1.2738k + kn) performs successfully on many real world examples.

A parameterized problem is a language L Ď Σ˚ ˆ Σ˚ over some finite al-phabet Σ. The instances of L are tuples xI, ky, where k is called the parameter.The parameter is often a non-negative integer, but it can be anything, e.g. arational number, three integers or a graph. For simplicity, we first assumethe parameter is a non-negative integer, i.e. L Ď Σ˚ ˆZ0. A parameterizedproblem is fixed-parameter tractable (fpt) if there exists an algorithm that solvesevery instance xI, ky of size n = |I| in time f (k) ¨ nc where f is an arbitrarycomputable function and c is a constant independent of both n and k. FPTis the class of all fixed-parameter tractable decision problems. In contrast toclassical tractability, some exponentiality is allowed, but confined to the pa-rameter only, thus better reflecting reality. For instance, the VERTEX COVER

problem parameterized by the size of the vertex cover, k, is in FPT. Note that1.2738k + kn ď 1.2738kn + kn = (1.2738k + k)n = f (k)n.

Hardness for parameterized classes is proven in the usual way, but usingfpt reductions instead of ordinary polynomial-time reductions. An fpt reduc-tion from a parameterized language L Ď Σ˚ ˆZ0 to another parameterizedlanguage L1 Ď Σ˚ ˆZ0 is a mapping R : Σ˚ ˆZ0 Ñ Σ˚ ˆZ0 such that:(1) xI, ky P L if and only if xI1, k1y = R(I, k) P L1; (2) there is a computablefunction f and a constant c such that R can be computed in time f (k) ¨ nc,where n = |I|; and (3) there is a computable function g such that k1 ď g(k).

Parameterized complexity offers a completeness theory, similar to thetheory of NP-completeness. It contains the following hierarchy of param-eterized complexity classes

FPT Ď W[1] Ď W[2] Ď W[3] Ď ¨ ¨ ¨ Ď W[P],

known as the W hierarchy. W[i] is defined using WEIGHTED CIRCUIT SATIS-FIABILITY problem: a parameterized problem is in W[i] if its every instancexI, ky is fpt-reducible to a boolean circuit of weft i, such that xI, ky has a so-lution if and only if the circuit has a truth assignment of weight (number

12

2. Parameterized Complexity

FPT

W[1]

W[2]

..

.W[P] para-NPXP

Figure 2.1: Parameterized Complexity Classes

of 1s in the input) at most k. Weft is the maximum number of gates withfan-in more than two on any path from the inputs to the output. The classW[P] corresponds to arbitrary circuits. For instance, the DOMINATING SET

problem parameterized by the size of the dominating set is W[2]-complete[25]. There are also classes outside of the W-hierarchy; para-NP is the classof problems solvable in non-deterministic time f (k)nc and XP is the class ofproblems solvable in time n f (k); where f is an arbitrary computable functionand c is a constant independent of both n and k. It is known that W[P] isa subset of both para-NP and XP [30]. Figure 2.1 shows the known relationbetween the mentioned parameterized complexity classes.

It is known that P Ď FPT and NP Ď para-NP, but otherwise thereis not much known about the relations between parameterized complexityclasses and the classical ones. For instance, there are NP-complete problemsthat are W[P]-complete and there are PSPACE-complete problems that are inFPT. The hardness of a parameterized problem depends a lot on the cho-sen parameter(s). For instance, consider the NP-complete problem GRAPH

k-COLORING: this problem is para-NP-hard if parameterized by k, the num-ber of available colors (mentioned in Paper 5 of this thesis) while it is in FPTif parameterized by n, the number of vertices (using a brute-force algorithm).

13

3 Automated Planning

1 What is Planning?

Planning is the art and practice ofthinking before acting.

Patrik Haslum

In this section we give a high-level description of planning. Imagine aworld or a system, like a car, a robotic spacecraft or a classroom with all thestudents and the teacher in it. There are some actions that change the state ofthe world. By executing the actions we can move between different states ofthe world. For instance, entering and leaving a student, giving a lecture ortaking a break are all actions that change the state of the classroom. Planningmay be defined as the problem of finding a sequence of such actions that leadto a specified goal state from a given initial state. A more abstract definitionis given by Ghallab, Nau, and Traverso [34]:

Planning is the reasoning side of acting. It is an abstract, explicitdeliberation process that chooses and organizes actions by antici-pating their expected outcomes. This deliberation aims at achiev-ing as best as possible some pre-stated objectives. Automatedplanning is an area of Artificial Intelligence (AI) that studies thisdeliberation process computationally.

We will continue with an example, Blocks world, which is one of the mostfamous planning domains in Artificial Intelligence.

15

3. AUTOMATED PLANNING

BA

C

C

B

A

Table

Figure 3.1: A Blocks world example; initial state (left) and goal state (right)

Example: Blocks world

In the Blocks world domain, we are given a set of identical blocks with dif-ferent letter identifiers on them. The blocks are positioned in several stackson a table. We have an arm to move the blocks. This arm can only pickblocks from the top of the stacks and put them on top of another stack or onthe table. The goal is to reach another specified configuration. This versionof Blocks world is known as the Elementary Blocks World (EBW). There arealso other variations of this problem [38].

To define the state of the world, we use the following predicate types:

• ON(A,B): Block A is on block B.

• ONTABLE(A): Block A is on the table.

• CLEAR(A): Block A has nothing on it.

• HOLDING(A): The arm holds block A.

• ARMEMPTY: The arm holds nothing.

We also define the following actions:

• UNSTACK(A,B): Pick up clear block A from block B.

• STACK(A,B): Place block A using the arm onto clear block B.

• PICKUP(A): Lift clear block A with the empty arm from the table.

• PUTDOWN(A): Place the held block A onto a free space on the table.

Figure 3.1 shows an example with three blocks. In this example, the initial(I) and goal (G) states contain the following predicates

I = tCLEAR(C), CLEAR(B), ONTABLE(A), ONTABLE(B), ON(C,A), ARMEMPTYu

G = tCLEAR(A), ONTABLE(C), ON(A,B), ON(B,C), ARMEMPTYu

16

1. What is Planning?

UNSTACK(A,B):

precondition list = tON(A,B), CLEAR(A), ARMEMPTYu

add list = tCLEAR(B), HOLDING(A)u

delete list = tON(A,B), ARMEMPTYu

STACK(A,B):

precondition list = tHOLDING(A), CLEAR(B)u

add list = tARMEMPTY, ON(A,B)u

delete list = tHOLDING(A), CLEAR(B)u

PICKUP(A):

precondition list = tCLEAR(A), ONTABLE(A), ARMEMPTYu

add list = tHOLDING(A)u

delete list = tONTABLE(A), ARMEMPTYu

PUTDOWN(A):

precondition list = tHOLDING(A)u

add list = tONTABLE(A), ARMEMPTYu

delete list = tHOLDING(A)u

Figure 3.2: The actions of the Blocks world domain

Every action is defined by three lists of predicates; a precondition list, thelist of predicates that need to hold for the action to be applicable in a state, anadd list, the list of predicates that are added after the action is executed in astate, and a delete list, the list of predicates that are deleted after the executionof the action. The actions are defined in Figure 3.2.

The objective is to find the shortest sequence of actions (a plan) that bringthe initial state (I) to the goal state (G). In this example, the unique optimalplan, ω, is the following list of actions:

ω =(

UNSTACK(C,A), PUTDOWN(C), PICKUP(B),

STACK(B,C), PICKUP(A), STACK(A,B))

˝

Given a Blocks world example with n blocks, one can produce in timeO(n) a plan (if one exists) that moves no block more that twice. However,the problem of finding the optimal plan is NP-hard [38].

Planning is closely connected to search. The state space of the blocksworld example can be seen as a directed graph where states (combination ofpredicates) are the nodes and there is an edge from node u to node v if thereis an action that transforms state u to state v. Hence, the problem of findingan optimal plan becomes the problem of finding the shortest path betweentwo nodes. Although this may seem easy using shortest path algorithms, themain challenge is that the number of nodes of this graph is exponential in

17


the input size. For instance, a blocks world instance with n blocks has morethan 2n2

states, which means even for n = 10 and checking 2 billion statesper second, a linear search takes more than 10 billion years. However, theproblem can easily be solved by a human in a few seconds.

2 Planning Model and Formalism

There are different ways to model the planning world. The differences comefrom how the following questions (motivated by Ghallab, Nau, and Traverso[34]) are answered:

• Is the planning process offline (static) or is it online (dynamic)?

• Is the state of the plan fully, partially or not observable?

• Is there only a single agent or are there several agents to execute ac-tions?

• Do variables have discrete or continuous domain? Do they have finiteor infinite domain?

• Are the actions deterministic, non-deterministic or stochastic?

• Do the actions have cost or duration?

• Can the actions be executed at the same time or do we only have se-quential plans?

• Is the objective to satisfy a goal or to find the optimal plan under somecriteria?

The most simple planning model is called classical planning. In classicalplanning the following assumptions are imposed:

• planning is static

• variable domains are discrete and finite

• actions are deterministic

• states are fully observable

• there is a single agent (sequential plan)

• actions have no cost or duration

There are many models and representations for planning in the literature[34]. Two common formalisms that are used to study the complexity of plan-ning are STRIPS and SAS+. This is because they are simple to work with andat the same time expressive enough to be able to represent many complexplanning domains.

18

2. Planning Model and Formalism

STRIPS

STRIPS (Stanford Research Institute Problem Solver) was defined by Fikesand Nilsson [29]. It is a planner and a formalism for describing planninginstance that is based on first-order logic: the state of the world is viewedas a set of ground and function-free first-order literals. We will not go intothe details, but merely note that the Blocks world example in this chapteris an example of a STRIPS planning instance. In many cases, one is contentwith the simplified variant of STRIPS where the state of the world is viewedas a set of propositional literals. This variant is referred to as propositionalSTRIPS.

SAS+

SAS+ (Simplified Action Structures) is very similar to propositional STRIPS,with the difference that the two-valued propositions are replaced by vari-ables that take their values from a finite domain. It was introduced by Bäck-ström and Klein [10]. Another difference is that instead of predicates thedomain values are assignments to variables in preconditions, effects, initialand goal state.

We have mostly used SAS+ formalism in the papers of this thesis andwill formally define it here (Although depending on the application, SAS+

is defined in slightly different ways in the papers of this Thesis.). A SAS+

planning instance is a 4-tuple xV, A, I, Gy, where

• V = tv1, v2, . . . , vnu is a set of variables over some finite domain D (insome definitions each variable v has an individual domain D(v)). De-fineD+ = DYtuu, where u is an extra value that stands for undefined.Dn is the set of total states whereDn

+ is the set of partial states. The valueof variable v in state s (partial or total) is denoted by s[v].

• A is a set of actions and each action a P A is of the form a : pre(a) Ñeff(a), where pre(a), eff(a) P Dn

+ and are called precondition and ef-fects (or post-conditions), respectively. An action a is applicable in totalstate s if and only if every value with a defined value (‰ u) in pre(a)has the same value in s. If a is applicable in s, the result of a in s is a to-tal state t where t[v] = s[v] if eff(a)[v] is undefined and t[v] = eff(a)[v]otherwise.

• I P Dn is the initial state.

• G P Dn+ is the goal state.

For two total states sI and sG, a sequence of actions (a1, a2, . . . , ak) is called aplan from sI to sG if and only if there exist total states ts0, s1, . . . , sku such thats0 = sI , sk = sG, ai is applicable in si´1 and si is the result of ai in si´1. Aplan from I to a total state s is a solution plan if s[v] = G[v] for every definedvariable in G.

19


3 Planning Problems

There are many planning problems described in the literature, with respectto different planning models, formalisms, constraints and objectives. In thisthesis we investigate the following problems:

PSAT Plan satisfiability problem (also known as plan existence problem(PE)) is the most basic planning problem. It asks if a planning instance(e.g. a SAS+ instance) has a solution.

LOP Length-optimal planning takes an integer ` and asks if the planninginstance has a solution of length at most `.

COP Cost-optimal planning is a generalization of LOP. It assigns a cost toeach action (c : A Ñ D), takes a cost threshold w and asks if there isplan of cost at most w.

PSP Partial-satisfaction problem is a generalization of PSAT. It takes an in-teger k in the input and asks if there is a plan that satisfies at least k ofthe goals.

NBP Net-benefit planning assigns a utility to each goal (u : G Ñ D) and acost to each action. It takes a net-benefit threshold b and asks if there isa plan with net-benefit at least b. The net-benefit of a plan is the sum ofutilities of the satisfied goals minus the plan cost.

We also investigate some problems beyond the boundaries of classical plan-ning. To continue, we first need to define partial-order and parallel plans. Fora more detailed explanation please refer to Paper 5.

Let Π be a planning problem instance. A partial-order plan for Π is atuple P = xA,ăy, where A is a set of action occurrences and ă is a partialorder on A. P is a valid plan for Π if all linearizations of it are valid plans. Thesize of an order is the number of relation tuples in it. A parallel plan is a tupleP = xA,ă, #y where xA,ăy is a partial-order plan and # is an irreflexive andsymmetric relation on A. The relation # is called the non-concurrency relation.For every two actions a and b, a#b indicates that they cannot be executedin parallel. Let d : A ÞÑ N denote the duration of each action. A parallelexecution of a parallel plan P is a function r : A ÞÑ N, denoting the releasetime for every action in A, such that: 1) If a ă b then r(a) + d(a) ď r(b), and2) If a#b then either r(a) + d(a) ď r(b) or r(b) + d(b) ď r(a). The length ofa parallel execution or maxaPAtr(a) + d(a)u denotes the latest finishing timeof any action. A k-processor parallel execution of a parallel plan is a parallelexecution such that for each time t P N there are at most k actions a such thatr(a) ď t ă r(a) + d(a).

PPL Parallel plan length takes a parallel plan P = xA,ă, #y, a length thresh-old ` and asks if there is a parallel execution for P of length at most `.

20

3. Planning Problems

PPLk This problem takes a parallel plan P = xA,ă, #y, an integer k, a lengththreshold ` and asks if there is a k-processor parallel execution for P oflength at most `.

MCR Minimum constrained reordering (MCR), takes a partial-order plan, avalue k and asks if there is a another valid order for the plan with size k.

MCD Minimum constrained deordering (MCD) is similar to MCR but re-quires the new order to be a subset of the initial order.

MPR Minimum parallel reordering asks if a partial-order plan has a reorder-ing of parallel length `.

MPD Minimum parallel deordering is the deordering version of MPR.

Complexity of Planning

Planning problems are known to be computationally hard in the general case.PSAT for propositional STRIPS and SAS+ is PSPACE-complete [11, 18] andfor STRIPS with predicates over infinite domains is undecidable [21], i.e. itis impossible to construct an algorithm that always generates a correct “yes”or “no” answer.

One method to reduce the computational complexity of planning is to ap-ply restrictions on the planning instance. The complexity of STRIPS planningvaries from constant time to undecidable depending on which restrictionswe make. These restrictions can be syntactic restrictions, such as restrictionson the number of preconditions and effects [2, 9, 18], or restrictions on ac-tion types [7, 10, 11, 36, 49]; semantic restrictions, such as restrictions on thevariable dependencies [42, 43, 46, 48, 49].

Finding new tractable problems in planning has received a special atten-tion not only because it helps to distinguish between easy and hard plan-ning instances, but also because one can use the polynomial-time solvableinstances in order to construct better heuristic functions and improve plan-ners [42, 43]. Many tractable classes of planning problems have been identi-fied [3, 11, 15, 36, 49, 52].

A recent method to tackle the hardness of planning problems is to get amore fine-grained complexity analysis by using parameterized complexity.Downey, Fellows, and Stege [26] showed that STRIPS planning is W[1]-hardparameterized by the plan length (Later, Bäckström, Jonsson, Ordyniak, andSzeider [9] strengthened this result and showed that LOP parameterized bythe plan length is W[2]-complete). Since then, the parameterized complexityof PSAT, LOP, COP and NBP has been studied under various restrictions anddifferent parameterizations [1, 6, 9, 12, 39, 56].

There are also other complexity results on variants of planning that areout of the scope of this thesis [13, 19, 28, 37, 41, 57, 58, 67].

21

4 Background andContributions

This section highlights the background and contributions of each paper inthis thesis.

1 Paper 1: Oversubscription Planning

Meysam Aghighi and Peter JonssonOversubscription planning: Complexity and compilability

In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence,Québec City, Québec, Canada, July 27 -31, 2014.

In classical planning all goals need to be satisfied at the same time. How-ever, in many real-world applications there are a larger number of possi-ble goals and the aim is to find a feasible subset of them [71]. This type ofplanning is called partial satisfaction or oversubscription planning. Severalcustom-made planners and heuristics have been suggested for oversubscrip-tion planning [14, 17, 61].

In this paper we study two different problems: the partial satisfactionproblem (PSP) where the goal is to maximize the number of achieved goals,and the net-benefit problem (NBP) where the goals is to maximize the totalweight of the achieved goals minus the plan cost. The net-benefit problemwas first formalized by Briel, Nigenda, Do, and Kambhampati [17] and wasshown to be PSPACE-complete. After that, there has not been a more detailedcomplexity analysis of this problem until paper 1 of this thesis. We considertwo types of restrictions in our complexity analysis: (1) restrictions on the

23

4. BACKGROUND AND CONTRIBUTIONS

number of preconditions and effects that were first proposed by Bylander[18], and (2) the PUBS restrictions proposed by Bäckström and Klein [10].

The results indicate that there is a high correlation between PSAT, COP,PSP and NBP. For instance, if PSAT for some set of instances X (under mildadditional assumptions) is NP-complete, then PSP for X is NP-complete, too.We also propose a polynomial-time reduction from NBP into PSAT, using anovel counter to keep track of plan cost during the plan execution and calcu-late the net-benefit after plan execution. The importance of this compilationis that the number of variables increase slowly in the size of the original in-stance, and this means that the algorithms for PSAT are able to solve NBP

instances with a limited slow-down. Compiling different planning problemsinto each other [55, 64, 72] and into other problems [16, 20, 54] have beenactive research areas for quite some time.

At the end of the paper we suggested a continuation of the work byimposing restrictions on the causal graph. Later, Katz and Mirkis [53] dis-covered several tractable fragments of net-benefit planning by restricting thecausal graph.

2 Paper 2: Causal Graphs for Tractability

Meysam Aghighi, Peter Jonsson, and Simon StåhlbergTractable cost-optimal planning over restricted polytree causal graphs

In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence,Austin, Texas, USA, January 25-30, 2015.

A causal graph is a directed graph with the domain variables as nodes.Node x in the graph is connected to node y if there exists some action a whereeff(a)[y] ‰ u and either eff(a)[x] ‰ u or pre(a)[x] ‰ u. There has been manystudies on the causal graph of a planning problem [15, 36, 51]. Giménez andJonsson [35] showed that plan generation is NP-hard for instances whosecausal graph is a directed path and have domain size at least 5. Katz andDomshlak [48, 50] have studied planning instances whose causal graphs areforks and inverted-forks. Katz and Keyder [52] have shown that cost-optimalplanning instances with bounded domain size and hourglass causal graphsare in P. In this paper, we generalize this result and prove that cost-optimalplanning instances with bounded domain size and polytree causal graphswith bounded diameter, are in P.

The algorithm in this paper is built on a novel notion of isomorphic vari-ables. Intuitively, two variables are isomorphic if the structure of their do-main transition graphs are isomorphic with regard to the initial value, goalvalue and actions. This results in the same behavior of these variables in ev-ery optimal solution, which allows us to combine them into a single variable.By repeating this process, we gradually reduce the size of the initial instanceand then solve it in polynomial time.

Brafman and Domshlak [15] have presented a polynomial-time algo-rithm for instances with binary variables and where the causal graph is a

24

3. Papers 3, 4: The Effect of Cost Domain in Cost-optimal and Net-benefitPlanning

polytree such that the in-degree of every vertex is bounded by a constant.Giménez and Jonsson [36] later generalized this result by giving an algorithmthat, instead of restricting the in-degree, restricts the number of prevail-conditions (a bound on the in-degree implies a bound on the number ofprevail-conditions but not the other way round). Our algorithm works dif-ferently: it allows not only binary domains, but any domain size boundedby a constant and it allows the actions to have an unbounded number ofprevail-conditions. However, this flexibility comes at the cost of boundingthe diameter by a constant.

3 Papers 3, 4: The Effect of Cost Domain in Cost-optimaland Net-benefit Planning

Meysam Aghighi and Christer BäckströmCost-optimal and net-benefit planning - A parameterised complexity view

In Proceedings of the Twenty-Fourth International Joint Conference on ArtificialIntelligence, IJCAI 2015, Buenos Aires, Argentina, July 25-31, 2015.

Meysam Aghighi and Christer BäckströmA multi-parameter complexity analysis of cost-optimal and net-benefit

planningIn Proceedings of the Twenty-Sixth International Conference on Automated

Planning and Scheduling, ICAPS 2016, London, UK, June 12-17, 2016.

Although cost-optimal and net-benefit planning have been vastly studiedin the literature, very little attention has been paid to motivate and discussthe choice of numeric domain for the action costs. For instance, Katz andDomshlak [49] and Helmert, Haslum, Hoffmann, and Nissim [44] use non-negative reals; Bäckström and Jonsson [8] and Yang, Culberson, Holte, Za-havi, and Felner [77] use non-negative integers; Cooper, Roquemaurel, andRégnier [24] use positive integers; Coles, Fox, Long, and Smith [23] use non-negative costs with no specified type; and Thayer, Stern, Felner, and Ruml[73] disregard cost types.

A traditional computational complexity analysis does not differentiatebetween different cost types; planning remains PSPACE-complete regardlessof whether action costs are positive or non-negative, integer or rational. Inpaper 3, we use parameterized complexity to draw a line between differentcost domains: cost-optimal planning is W[2]-complete for positive integer(Z+) costs while it is para-NP-hard if the costs are non-negative integers (Z0)or positive rationals (Q+). This is a strong indication that the latter cases aresubstantially harder. The paper continues by providing a detailed complex-ity map of both cost-optimal and net-benefit planning problems, with PUBSrestrictions and restrictions on the number of preconditions and effects.

This result may seem in contradiction with the usual assumption thatthe use of rational costs is the same as integer costs: by multiplying all ac-tion costs and plan cost threshold by a sufficiently large number. By using

25

4. BACKGROUND AND CONTRIBUTIONS

this transformation, from any rational-cost planning instance we can get aninteger-cost instance and solve it by using an integer-cost planner. Althoughthis method is correct and the new instance with integer costs has a solution ifand only if the initial rational-cost instance has a solution, the transformationdoes not preserve the parameterized complexity (it is not an fpt-reduction).

The work is continued by a multi-parameter complexity analysis of thesame problems in paper 4. In this work we consider a total of thirteen param-eters including the plan length, `, the plan cost, k, the largest denominatorof the action costs, d, and the sum of variable utilities, t. The latter parame-ter only applies to NBP. Two new cost domains were also considered in thispaper, the non-negative rationals (Q0) and rationals greater than or equal toone (Q1). Our results show that COP remains W[2]-hard for all cost domainseven when combining all parameters. We also show the W[2]-membershipof COP for the following parameter combinations and domains:

• tk, `u for Z0

• tk, du for Q+

• tk, d, `u for Q0

Another interesting and important result of paper 4 is the effect of param-eter t, the sum of all utilities. We show, through an fpt reduction, that NBP

parameterized by t is reducible to COP parameterized by k over the same nu-meric domain. This reduction is both related to the method used by Keyderand Geffner [55] to compile away soft goals, and to the compilation of NBP

to PSAT by Aghighi and Jonsson [2] in paper 1 of this thesis. However, if weuse the second method and apply the counter reduction from paper 1, wehave to limit ourselves only to integer costs.

4 Paper 5: Plan Optimization

Meysam Aghighi and Christer BäckströmPlan reordering and parallel execution – a parameterized complexity viewIn Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, San

Francisco, California, USA, February 4-9, 2017.

Some real-world applications require a more flexible structure than a se-quential plan. For instance the applications that involve real-time executionor multi-agent taskability [75, 76]. One method to model this flexibility isto use partial-order plans. This allows not only different ways to linearize aplan, but also the possibility to postpone making decisions for linearizationto run-time. Another important feature of a partial-order plan is its ability tobe used as a parallel plan, where mutually unordered actions can be executedin parallel. Some planners directly produce a partial-order plan. At the sametime, there has been attempts to produce a partial-order plan from the outputof a total-order planner. The core idea is to remove the unnecessary order-ing relations and several algorithms have been proposed to do this, both in

26

4. Paper 5: Plan Optimization

the early 1990s and recently [47, 62, 63, 66, 74]. These algorithms have beenapplied to various planning domains including workflow planning [68], au-tomatic generation of narratives [40], composition of web services [59], planexecution and monitoring [62], plan repair [69], and distributed control sys-tems [65].

Bäckström [5] theoretically approached the problem and defined deorder-ing and reordering of a partial-order plan. Deordering of a plan is to replacethe plan order by a subset of the order such that the new plan remains avalid partial-order plan. Reordering is to replace the plan order with anyother order such that the result is a valid partial-order plan. He showed thatit is NP-complete to optimize the size (number of relation tuples) of both de-ordering and reordering. There can be other criteria for plan optimization aswell. One example is the length of the parallel execution of the plan. Bäck-ström [5] has also looked into this problem and showed that both finding theshortest parallel execution of a parallel plan (PPL) and optimization of theparallel length by reordering (MPR) are NP-complete problems.

In paper 5 of this thesis, we initiate a parameterized complexity investi-gation of the problems proposed by Bäckström [5] in order to shed more lighton them and get more refined results. Two important parameters of a partial-order are its width (the size of the largest anti-chain) and height (the lengthof the longest chain). We study the effect of these parameters on the parame-terized complexity of finding the optimal deordering (MCD) and reordering(MCR). We show that both MCD and MCR are W[2]-hard and in W[P] if pa-rameterized by the desired order size, and MCD is in FPT if parameterizedby the original order size.

We further show that the length of the parallel execution of a plan ishighly related to both the height of the partial order and the chromatic num-ber of the non-concurrency graph. Moreover, we prove that finding the min-imum parallel execution of a plan is in FPT if parameterized by the size ofthe non-concurrency relation.

Finally, we propose a more realistic criterion for plan optimization: thelength of the k-processor parallel execution. We study this criterion throughthe problem PPLk. We prove that PPLk is in FPT if parameterized by the com-bination of: (1) the size of the non-concurrency relation, (2) the length of thek-processor parallel execution and (3) the number of processor respectively.On the other hand, the problem remains para-NP-hard if only parameterizedby tn#, lpu; i.e. significantly harder than the case with unlimited number ofprocessors (PPL). This is consistent with the intuition that the bound on thenumber of processors makes the problem much harder.

27

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35

Papers

The articles associated with this thesis have been removed for copyright

reasons. For more details about these see:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-136280



Department of Computer and Information Science Linköpings universitet

Dissertations

Linköping Studies in Science and Technology

Linköping Studies in Arts and Science Linköping Studies in Statistics

Linköping Studies in Information Science

Linköping Studies in Science and Technology No 14 Anders Haraldsson: A Program Manipulation

System Based on Partial Evaluation, 1977, ISBN 91-7372-144-1.

No 17 Bengt Magnhagen: Probability Based Verification of Time Margins in Digital Designs, 1977, ISBN 91-7372-157-3.

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No 22 Jaak Urmi: A Machine Independent LISP Compiler and its Implications for Ideal Hardware, 1978, ISBN 91-7372-188-3.

No 33 Tore Risch: Compilation of Multiple File Queries in a Meta-Database System 1978, ISBN 91- 7372-232-4.

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No 55 Pär Emanuelson: Performance Enhancement in a Well-Structured Pattern Matcher through Partial Evaluation, 1980, ISBN 91-7372-403-3.

No 58 Bengt Johnsson, Bertil Andersson: The Human-Computer Interface in Commercial Systems, 1981, ISBN 91-7372-414-9.

No 69 H. Jan Komorowski: A Specification of an Abstract Prolog Machine and its Application to Partial Evaluation, 1981, ISBN 91-7372-479-3.

No 71 René Reboh: Knowledge Engineering Techniques and Tools for Expert Systems, 1981, ISBN 91-7372-489-0.

No 77 Östen Oskarsson: Mechanisms of Modifiability in large Software Systems, 1982, ISBN 91- 7372-527-7.

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No 170 Zebo Peng: A Formal Methodology for Automated Synthesis of VLSI Systems, 1987, ISBN 91-7870-225-9.

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No 192 Dimiter Driankov: Towards a Many Valued Logic of Quantified Belief, 1988, ISBN 91-7870-374-3.

No 213 Lin Padgham: Non-Monotonic Inheritance for an Object Oriented Knowledge Base, 1989, ISBN 91-7870-485-5.

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No 252 Peter Eklund: An Epistemic Approach to Interactive Design in Multiple Inheritance Hierarchies, 1991, ISBN 91-7870-784-6.

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No 260 Nahid Shahmehri: Generalized Algorithmic Debugging, 1991, ISBN 91-7870-828-1.

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No 302 Tingting Zhang: A Study in Diagnosis Using Classification and Defaults, 1993, ISBN 91-7871-078-2

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No 371 Bengt Savén: Business Models for Decision Support and Learning. A Study of Discrete-Event Manufacturing Simulation at Asea/ ABB 1968-1993, 1995, ISBN 91-7871-494-X.

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No 480 Mikael Lindvall: An Empirical Study of Require-ments-Driven Impact Analysis in Object-Oriented Software Evolution, 1997, ISBN 91-7871-927-5.

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No 567 Jörgen Lindström: Does Distance matter? On geo-graphical d ispersion in organisations, 1999, ISBN 91-7219-439-1.

No 582 Vanja Josifovski: Design, Implementation and Evaluation of a Distribu ted Mediator System for Data Integration, 1999, ISBN 91-7219-482-0.

No 589 Rita Kovordányi: Modeling and Simulating Inhibitory Mechanisms in Mental Image Reinterpretation - Towards Cooperative Human-Computer Creativity, 1999, ISBN 91-7219-506-1.

No 592 Mikael Ericsson: Supporting the Use of Design Knowledge - An Assessment of Commenting Agents, 1999, ISBN 91-7219-532-0.

No 593 Lars Karlsson: Actions, Interactions and Narratives, 1999, ISBN 91-7219-534-7.

No 594 C. G. Mikael Johansson: Social and Organizational Aspects of Requirements Engineering Methods - A practice-oriented approach, 1999, ISBN 91-7219-541-X.

No 595 Jörgen Hansson: Value-Driven Multi-Class Overload Management in Real-Time Database Systems, 1999, ISBN 91-7219-542-8.

No 596 Niklas Hallberg: Incorporating User Values in the Design of Information Systems and Services in the Public Sector: A Methods Approach, 1999, ISBN 91-7219-543-6.

No 597 Vivian Vimarlund: An Economic Perspective on the Analysis of Impacts of Information Technology: From Case Stud ies in Health -Care towards General Models and Theories, 1999, ISBN 91-7219-544-4.

No 598 Johan Jenvald: Methods and Tools in Computer-Supported Taskforce Training, 1999, ISBN 91-7219-547-9.

No 607 Magnus Merkel: Understanding and enhancing translation by parallel text processing, 1999, ISBN 91-7219-614-9.

No 611 Silvia Coradeschi: Anchoring symbols to sensory data, 1999, ISBN 91-7219-623-8.

No 613 Man Lin: Analysis and Synthesis of Reactive Systems: A Generic Layered Architecture Perspective, 1999, ISBN 91-7219-630-0.

No 618 Jimmy Tjäder: Systemimplementering i praktiken - En stud ie av logiker i fyra projekt, 1999, ISBN 91-7219-657-2.

No 627 Vadim Engelson: Tools for Design, Interactive Simulation, and Visualization of Object-Oriented Models in Scientific Computing, 2000, ISBN 91-7219-709-9.

No 637 Esa Falkenroth: Database Technology for Control and Simulation, 2000, ISBN 91-7219-766-8.

No 639 Per-Arne Persson: Bringing Power and Knowledge Together: Information Systems Design for Autonomy and Control in Command Work, 2000, ISBN 91-7219-796-X.

No 660 Erik Larsson: An Integrated System-Level Design for Testability Methodology, 2000, ISBN 91-7219-890-7.

No 688 Marcus Bjäreland: Model-based Execution Monitoring, 2001, ISBN 91-7373-016-5.

No 689 Joakim Gustafsson: Extending Temporal Action Logic, 2001, ISBN 91-7373-017-3.

No 720 Carl-Johan Petri: Organizational Information Provi-sion - Managing Mandatory and Discretionary Use of Information Technology, 2001, ISBN -91-7373-126-9.

No 724 Paul Scerri: Designing Agents for Systems with Ad -justable Autonomy, 2001, ISBN 91 7373 207 9.

No 725 Tim Heyer: Semantic Inspection of Software Artifacts: From Theory to Practice, 2001, ISBN 91 7373 208 7.

No 726 Pär Carlshamre: A Usability Perspective on Require-ments Engineering - From Methodology to Product Development, 2001, ISBN 91 7373 212 5.

No 732 Juha Takkinen: From Information Management to Task Management in Electronic Mail, 2002, ISBN 91 7373 258 3.

No 745 Johan Åberg: Live Help Systems: An Approach to Intelligent Help for Web Information Systems, 2002, ISBN 91-7373-311-3.

No 746 Rego Granlund: Monitoring Distributed Teamwork Training, 2002, ISBN 91-7373-312-1.

No 757 Henrik André-Jönsson: Indexing Strategies for Time Series Data, 2002, ISBN 917373-346-6.

No 747 Anneli Hagdahl: Development of IT-supported Interorganisational Collaboration - A Case Study in the Swedish Public Sector, 2002, ISBN 91-7373-314-8.

No 749 Sofie Pilemalm: Information Technology for Non-Profit Organisations - Extended Participatory Design of an Information System for Trade Union Shop Stewards, 2002, ISBN 91-7373-318-0.

No 765 Stefan Holmlid: Adapting users: Towards a theory of use quality, 2002, ISBN 91-7373-397-0.

No 771 Magnus Morin: Multimedia Representations of Dis-tributed Tactical Operations, 2002, ISBN 91-7373-421-7.

No 772 Pawel Pietrzak: A Type-Based Framework for Locat-ing Errors in Constraint Logic Programs, 2002, ISBN 91-7373-422-5.

No 758 Erik Berglund: Library Communication Among Pro-grammers Worldwide, 2002, ISBN 91-7373-349-0.

No 774 Choong-ho Yi: Modelling Object-Oriented Dynamic Systems Using a Logic-Based Framework, 2002, ISBN 91-7373-424-1.

No 779 Mathias Broxvall: A Study in the Computational Complexity of Temporal Reasoning, 2002, ISBN 91-7373-440-3.

No 793 Asmus Pandikow: A Generic Principle for Enabling Interoperability of Structured and Object-Oriented Analysis and Design Tools, 2002, ISBN 91-7373-479-9.

No 785 Lars Hult: Publika Informationstjänster. En stud ie av den Internetbaserade encyklopedins bruksegenska-per, 2003, ISBN 91-7373-461-6.

No 800 Lars Taxén: A Framework for the Coord ination of Complex Systems´ Development, 2003, ISBN 91-7373-604-X

No 808 Klas Gäre: Tre perspektiv på förväntningar och förändringar i samband med införande av informationssystem, 2003, ISBN 91-7373-618-X.

No 821 Mikael Kindborg: Concurrent Comics - programming of social agents by child ren, 2003, ISBN 91-7373-651-1.

No 823 Christina Ölvingson: On Development of Information Systems with GIS Functionality in Public Health Informatics: A Requirements Engineering Approach, 2003, ISBN 91-7373-656-2.

No 828 Tobias Ritzau: Memory Efficient Hard Real-Time Garbage Collection, 2003, ISBN 91-7373-666-X.

No 833 Paul Pop: Analysis and Synthesis of Communication-Intensive Heterogeneous Real-Time Systems, 2003, ISBN 91-7373-683-X.

No 852 Johan Moe: Observing the Dynamic Behaviour of Large Distributed Systems to Improve Development and Testing --- An Empirical Study in Software Engineering, 2003, ISBN 91-7373-779-8.

No 867 Erik Herzog: An Approach to Systems Engineering Tool Data Representation and Exchange, 2004, ISBN 91-7373-929-4.

No 872 Aseel Berglund: Augmenting the Remote Control: Stud ies in Complex Information Navigation for Digital TV, 2004, ISBN 91-7373-940-5.

No 869 Jo Skåmedal: Telecommuting’s Implications on Travel and Travel Patterns, 2004, ISBN 91-7373-935-9.

No 870 Linda Askenäs: The Roles of IT - Stud ies of Organising when Implementing and Using Enterprise Systems, 2004, ISBN 91-7373-936-7.

No 874 Annika Flycht-Eriksson: Design and Use of Ontolo-gies in Information-Provid ing Dialogue Systems, 2004, ISBN 91-7373-947-2.

No 873 Peter Bunus: Debugging Techniques for Equation-Based Languages, 2004, ISBN 91-7373-941-3.

No 876 Jonas Mellin: Resource-Pred ictable and Efficient Monitoring of Events, 2004, ISBN 91-7373-956-1.

No 883 Magnus Bång: Computing at the Speed of Paper: Ubiquitous Computing Environments for Healthcare Professionals, 2004, ISBN 91-7373-971-5

No 882 Robert Eklund: Disfluency in Swedish human-human and human-machine travel booking d i-alogues, 2004, ISBN 91-7373-966-9.

No 887 Anders Lindström: English and other Foreign Linguistic Elements in Spoken Swedish. Stud ies of Productive Processes and their Mod elling using Finite-State Tools, 2004, ISBN 91-7373-981-2.

No 889 Zhiping Wang: Capacity-Constrained Production-in-ventory systems - Modelling and Analysis in both a trad itional and an e-business context, 2004, ISBN 91-85295-08-6.

No 893 Pernilla Qvarfordt: Eyes on Multimodal Interaction, 2004, ISBN 91-85295-30-2.

No 910 Magnus Kald: In the Borderland between Strategy and Management Control - Theoretical Framework and Empirical Evidence, 2004, ISBN 91-85295-82-5.

No 918 Jonas Lundberg: Shaping Electronic News: Genre Perspectives on Interaction Design, 2004, ISBN 91-85297-14-3.

No 900 Mattias Arvola: Shades of use: The dynamics of interaction design for sociable use, 2004, ISBN 91-85295-42-6.

No 920 Luis Alejandro Cortés: Verification and Scheduling Techniques for Real-Time Embedded Systems, 2004, ISBN 91-85297-21-6.

No 929 Diana Szentivanyi: Performance Stud ies of Fault-Tolerant Middleware, 2005, ISBN 91-85297-58-5.

No 933 Mikael Cäker: Management Accounting as Constructing and Opposing Customer Focus: Three Case Stud ies on Management Accounting and Customer Relations, 2005, ISBN 91-85297-64-X.

No 937 Jonas Kvarnström: TALplanner and Other Extensions to Temporal Action Logic, 2005, ISBN 91-85297-75-5.

No 938 Bourhane Kadmiry: Fuzzy Gain-Scheduled Visual Servoing for Unmanned Helicopter, 2005, ISBN 91-85297-76-3.

No 945 Gert Jervan: Hybrid Built-In Self-Test and Test Generation Techniques for Digital Systems, 2005, ISBN: 91-85297-97-6.

No 946 Anders Arpteg: Intelligent Semi-Structured Informa-tion Extraction, 2005, ISBN 91-85297-98-4.

No 947 Ola Angelsmark: Constructing Algorithms for Con-straint Satisfaction and Related Problems - Methods and Applications, 2005, ISBN 91-85297-99-2.

No 963 Calin Curescu: Utility-based Optimisation of Resource Allocation for Wireless Networks, 2005, ISBN 91-85457-07-8.

No 972 Björn Johansson: Joint Control in Dynamic Situations, 2005, ISBN 91-85457-31-0.

No 974 Dan Lawesson: An Approach to Diagnosability Analysis for Interacting Finite State Systems, 2005, ISBN 91-85457-39-6.

No 979 Claudiu Duma: Security and Trust Mechanisms for Groups in Distributed Services, 2005, ISBN 91-85457-54-X.

No 983 Sorin Manolache: Analysis and Optimisation of Real-Time Systems with Stochastic Behaviour, 2005, ISBN 91-85457-60-4.

No 986 Yuxiao Zhao: Standard s-Based Application Integration for Business-to-Business Communications, 2005, ISBN 91-85457-66-3.

No 1004 Patrik Haslum: Admissible Heuristics for Automated Planning, 2006, ISBN 91-85497-28-2.

No 1005 Aleksandra Tešanovic: Developing Reusable and Reconfigurable Real-Time Software using Aspects and Components, 2006, ISBN 91-85497-29-0.

No 1008 David Dinka: Role, Identity and Work: Extending the design and development agenda, 2006, ISBN 91-85497-42-8.

No 1009 Iakov Nakhimovski: Contributions to the Modeling and Simulation of Mechanical Systems with Detailed Contact Analysis, 2006, ISBN 91-85497-43-X.

No 1013 Wilhelm Dahllöf: Exact Algorithms for Exact Satisfiability Problems, 2006, ISBN 91-85523-97-6.

No 1016 Levon Saldamli: PDEModelica - A High-Level Lan-guage for Modeling with Partial Differential Equa-tions, 2006, ISBN 91-85523-84-4.

No 1017 Daniel Karlsson: Verification of Component-based Embedded System Designs, 2006, ISBN 91-85523-79-8

No 1018 Ioan Chisalita: Communication and Networking Techniques for Traffic Safety Systems, 2006, ISBN 91-85523-77-1.

No 1019 Tarja Susi: The Puzzle of Social Activity - The Significance of Tools in Cognition and Cooperation, 2006, ISBN 91-85523-71-2.

No 1021 Andrzej Bednarski: Integrated Optimal Code Gener-ation for Digital Signal Processors, 2006, ISBN 91-85523-69-0.

No 1022 Peter Aronsson: Automatic Parallelization of Equa-tion-Based Simulation Programs, 2006, ISBN 91-85523-68-2.

No 1030 Robert Nilsson: A Mutation-based Framework for Automated Testing of Timeliness, 2006, ISBN 91-85523-35-6.

No 1034 Jon Edvardsson: Techniques for Automatic Generation of Tests from Programs and Specifications, 2006, ISBN 91-85523-31-3.

No 1035 Vaida Jakoniene: Integration of Biological Data, 2006, ISBN 91-85523-28-3.

No 1045 Genevieve Gorrell: Generalized Hebbian Algorithms for Dimensionality Reduction in Natural Language Processing, 2006, ISBN 91-85643-88-2.

No 1051 Yu-Hsing Huang: Having a New Pair of Glasses - Applying Systemic Accident Models on Road Safety, 2006, ISBN 91-85643-64-5.

No 1054 Åsa Hedenskog: Perceive those things which cannot be seen - A Cognitive Systems Engineering perspective on requirements management, 2006, ISBN 91-85643-57-2.

No 1061 Cécile Åberg: An Evaluation Platform for Semantic Web Technology, 2007, ISBN 91-85643-31-9.

No 1073 Mats Grindal: Handling Combinatorial Explosion in Software Testing, 2007, ISBN 978-91-85715-74-9.

No 1075 Almut Herzog: Usable Security Policies for Runtime Environments, 2007, ISBN 978-91-85715-65-7.

No 1079 Magnus Wahlström: Algorithms, measures, and upper bounds for Satisfiability and related problems, 2007, ISBN 978-91-85715-55-8.

No 1083 Jesper Andersson: Dynamic Software Architectures, 2007, ISBN 978-91-85715-46-6.

No 1086 Ulf Johansson: Obtaining Accurate and Compre-hensible Data Mining Models - An Evolu tionary Approach, 2007, ISBN 978-91-85715-34-3.

No 1089 Traian Pop: Analysis and Optimisation of Distributed Embedded Systems with Heterogeneous Scheduling Policies, 2007, ISBN 978-91-85715-27-5.

No 1091 Gustav Nordh: Complexity Dichotomies for CSP-related Problems, 2007, ISBN 978-91-85715-20-6.

No 1106 Per Ola Kristensson: Discrete and Continuous Shape Writing for Text Entry and Control, 2007, ISBN 978-91-85831-77-7.

No 1110 He Tan: Aligning Biomedical Ontologies, 2007, ISBN 978-91-85831-56-2.

No 1112 Jessica Lindblom: Minding the body - Interacting so-cially through embodied action, 2007, ISBN 978-91-85831-48-7.

No 1113 Pontus Wärnestål: Dialogue Behavior Management in Conversational Recommender Systems, 2007, ISBN 978-91-85831-47-0.

No 1120 Thomas Gustafsson: Management of Real-Time Data Consistency and Transient Overloads in Embedded Systems, 2007, ISBN 978-91-85831-33-3.

No 1127 Alexandru Andrei: Energy Efficient and Pred ictable Design of Real-time Embedded Systems, 2007, ISBN 978-91-85831-06-7.

No 1139 Per Wikberg: Eliciting Knowledge from Experts in Modeling of Complex Systems: Managing Variation and Interactions, 2007, ISBN 978-91-85895-66-3.

No 1143 Mehdi Amirijoo: QoS Control of Real-Time Data Services under Uncertain Workload , 2007, ISBN 978-91-85895-49-6.

No 1150 Sanny Syberfeldt: Optimistic Replication with For-ward Conflict Resolution in Distributed Real-Time Databases, 2007, ISBN 978-91-85895-27-4.

No 1155 Beatrice Alenljung: Envisioning a Future Decision Support System for Requirements Engineering - A Holistic and Human-centred Perspective, 2008, ISBN 978-91-85895-11-3.

No 1156 Artur Wilk: Types for XML with Application to Xcerpt, 2008, ISBN 978-91-85895-08-3.

No 1183 Adrian Pop: Integrated Model-Driven Development Environments for Equation-Based Object-Oriented Languages, 2008, ISBN 978-91-7393-895-2.

No 1185 Jörgen Skågeby: Gifting Technologies - Ethnographic Stud ies of End -users and Social Media Sharing, 2008, ISBN 978-91-7393-892-1.

No 1187 Imad-Eldin Ali Abugessaisa: Analytical tools and information-sharing methods supporting road safety organizations, 2008, ISBN 978-91-7393-887-7.

No 1204 H. Joe Steinhauer: A Representation Scheme for De-scription and Reconstruction of Object Configurations Based on Qualitative Relations, 2008, ISBN 978-91-7393-823-5.

No 1222 Anders Larsson: Test Optimization for Core-based System-on-Chip, 2008, ISBN 978-91-7393-768-9.

No 1238 Andreas Borg: Processes and Models for Capacity Requirements in Telecommunication Systems, 2009, ISBN 978-91-7393-700-9.

No 1240 Fredrik Heintz: DyKnow: A Stream-Based Know-ledge Processing Middleware Framework, 2009, ISBN 978-91-7393-696-5.

No 1241 Birgitta Lindström: Testability of Dynamic Real-Time Systems, 2009, ISBN 978-91-7393-695-8.

No 1244 Eva Blomqvist: Semi-automatic Ontology Construc-tion based on Patterns, 2009, ISBN 978-91-7393-683-5.

No 1249 Rogier Woltjer: Functional Modeling of Constraint Management in Aviation Safety and Command and Control, 2009, ISBN 978-91-7393-659-0.

No 1260 Gianpaolo Conte: Vision-Based Localization and Guidance for Unmanned Aerial Vehicles, 2009, ISBN 978-91-7393-603-3.

No 1262 AnnMarie Ericsson: Enabling Tool Support for For-mal Analysis of ECA Rules, 2009, ISBN 978-91-7393-598-2.

No 1266 Jiri Trnka: Exploring Tactical Command and Control: A Role-Playing Simulation Approach, 2009, ISBN 978-91-7393-571-5.

No 1268 Bahlol Rahimi: Supporting Collaborative Work through ICT - How End-users Think of and Adopt Integrated Health Information Systems, 2009, ISBN 978-91-7393-550-0.

No 1274 Fredrik Kuivinen: Algorithms and Hardness Results for Some Valued CSPs, 2009, ISBN 978-91-7393-525-8.

No 1281 Gunnar Mathiason: Virtual Full Replication for Scalable Distributed Real-Time Databases, 2009, ISBN 978-91-7393-503-6.

No 1290 Viacheslav Izosimov: Scheduling and Optimization of Fault-Tolerant Distribu ted Embedded Systems, 2009, ISBN 978-91-7393-482-4.

No 1294 Johan Thapper: Aspects of a Constraint Optimisation Problem, 2010, ISBN 978-91-7393-464-0.

No 1306 Susanna Nilsson: Augmentation in the Wild : User Centered Development and Evaluation of Augmented Reality Applications, 2010, ISBN 978-91-7393-416-9.

No 1313 Christer Thörn: On the Quality of Feature Models, 2010, ISBN 978-91-7393-394-0.

No 1321 Zhiyuan He: Temperature Aware and Defect-Probability Driven Test Scheduling for System-on-Chip, 2010, ISBN 978-91-7393-378-0.

No 1333 David Broman: Meta-Languages and Semantics for Equation-Based Modeling and Simulation, 2010, ISBN 978-91-7393-335-3.

No 1337 Alexander Siemers: Contributions to Modelling and Visualisation of Multibody Systems Simulations with

Detailed Contact Analysis, 2010, ISBN 978-91-7393-317-9.

No 1354 Mikael Asplund: Disconnected Discoveries: Availability Stud ies in Partitioned Networks, 2010, ISBN 978-91-7393-278-3.

No 1359 Jana Rambusch: Mind Games Extended: Understanding Gameplay as Situated Activity, 2010, ISBN 978-91-7393-252-3.

No 1373 Sonia Sangari: Head Movement Correlates to Focus Assignment in Swedish,2011,ISBN 978-91-7393-154-0.

No 1374 Jan-Erik Källhammer: Using False Alarms when Developing Automotive Active Safety Systems, 2011, ISBN 978-91-7393-153-3.

No 1375 Mattias Eriksson: Integrated Code Generation, 2011, ISBN 978-91-7393-147-2.

No 1381 Ola Leifler: Affordances and Constraints of Intelligent Decision Support for Military Command and Control --- Three Case Stud ies of Support Systems, 2011, ISBN 978-91-7393-133-5.

No 1386 Soheil Samii: Quality-Driven Synthesis and Optimization of Embedded Control Systems, 2011, ISBN 978-91-7393-102-1.

No 1419 Erik Kuiper: Geographic Routing in Intermittently-connected Mobile Ad Hoc Networks: Algorithms and Performance Models, 2012, ISBN 978-91-7519-981-8.

No 1451 Sara Stymne: Text Harmonization Strategies for Phrase-Based Statistical Machine Translation, 2012, ISBN 978-91-7519-887-3.

No 1455 Alberto Montebelli: Modeling the Role of Energy Management in Embodied Cognition, 2012, ISBN 978-91-7519-882-8.

No 1465 Mohammad Saifullah: Biologically-Based Interactive Neural Network Models for Visual Attention and Object Recognition, 2012, ISBN 978-91-7519-838-5.

No 1490 Tomas Bengtsson: Testing and Logic Optimization Techniques for Systems on Chip, 2012, ISBN 978-91-7519-742-5.

No 1481 David Byers: Improving Software Security by Preventing Known Vulnerabilities, 2012, ISBN 978-91-7519-784-5.

No 1496 Tommy Färnqvist: Exploiting Structure in CSP-related Problems, 2013, ISBN 978-91-7519-711-1.

No 1503 John Wilander: Contributions to Specification, Implementation, and Execution of Secure Software, 2013, ISBN 978-91-7519-681-7.

No 1506 Magnus Ingmarsson: Creating and Enabling the Useful Service Discovery Experience, 2013, ISBN 978-91-7519-662-6.

No 1547 Wladimir Schamai: Model-Based Verification of Dynamic System Behavior against Requirements: Method , Language, and Tool, 2013, ISBN 978-91-7519-505-6.

No 1551 Henrik Svensson: Simulations, 2013, ISBN 978-91-7519-491-2.

No 1559 Sergiu Rafiliu: Stability of Adaptive Distribu ted Real-Time Systems with Dynamic Resource Management, 2013, ISBN 978-91-7519-471-4.

No 1581 Usman Dastgeer: Performance-aware Component Composition for GPU-based Systems, 2014, ISBN 978-91-7519-383-0.

No 1602 Cai Li: Reinforcement Learning of Locomotion based on Central Pattern Generators, 2014, ISBN 978-91-7519-313-7.

No 1652 Roland Samlaus: An Integrated Development Environment with Enhanced Domain -Specific

Interactive Model Validation, 2015, ISBN 978-91-7519-090-7.

No 1663 Hannes Uppman: On Some Combinatorial Optimization Problems: Algorithms and Complexity, 2015, ISBN 978-91-7519-072-3.

No 1664 Martin Sjölund: Tools and Methods for Analysis, Debugging, and Performance Improvement of Equation-Based Models, 2015, ISBN 978-91-7519-071-6.

No 1666 Kristian Stavåker: Contribu tions to Simulation of Modelica Models on Data-Parallel Multi-Core Architectures, 2015, ISBN 978-91-7519-068-6.

No 1680 Adrian Lifa: Hardware/ Software Codesign of Embedded Systems with Reconfigurable and Heterogeneous Platforms, 2015, ISBN 978-91-7519-040-2.

No 1685 Bogdan Tanasa: Timing Analysis of Distributed Embedded Systems with Stochastic Workload and Reliability Constraints, 2015, ISBN 978-91-7519-022-8.

No 1691 Håkan Warnquist: Troubleshooting Trucks --- Automated Planning and Diagnosis, 2015, ISBN 978-91-7685-993-3.

No 1702 Nima Aghaee: Thermal Issues in Testing of Advanced Systems on Chip, 2015, ISBN 978-91-7685-949-0.

No 1715 Maria Vasilevskaya: Security in Embedded Systems: A Model-Based Approach with Risk Metrics, 2015, ISBN 978-91-7685-917-9.

No 1729 Ke Jiang: Security-Driven Design of Real-Time Embedded System, 2016, ISBN 978-91-7685-884-4.

No 1733 Victor Lagerkvist: Strong Partial Clones and the Complexity of Constraint Satisfaction Problems: Limitations and Applications, 2016, ISBN 978-91-7685-856-1.

No 1734 Chandan Roy: An Informed System Development Approach to Tropical Cyclone Track and Intensity Forecasting, 2016, ISBN 978-91-7685-854-7.

No 1746 Amir Aminifar: Analysis, Design, and Optimization of Embedded Control Systems, 2016, ISBN 978-91-7685-826-4.

No 1747 Ekhiotz Vergara: Energy Modelling and Fairness for Efficient Mobile Communication , 2016, ISBN 978-91-7685-822-6.

No 1748 Dag Sonntag: Chain Graphs --- Interpretations, Expressiveness and Learning Algorithms , 2016, ISBN 978-91-7685-818-9.

No 1768 Anna Vapen: Web Authentication using Third -Parties in Untrusted Environments , 2016, ISBN 978-91-7685-753-3.

No 1778 Magnus Jandinger: On a Need to Know Basis: A Conceptual and Methodological Framework for Modelling and Analysis of Information Demand in an Enterprise Context , 2016, ISBN 978-91-7685-713-7.

No 1798 Rahul Hiran: Collaborative Network Security: Targeting Wide-area Routing and Edge-network Attacks, 2016, ISBN 978-91-7685-662-8.

No 1813 Nicolas Melot: Algorithms and Framework for Energy Efficient Parallel Stream Computing on Many-Core Architectures, 2016, ISBN 978-91-7685-623-9.

No 1823 Amy Rankin: Making Sense of Adaptations: Resilience in High-Risk Work, 2017, ISBN 978-91-7685-596-6.

No 1831 Lisa Malmberg: Build ing Design Capability in the Public Sector: Expand ing the Horizons of Development, 2017, ISBN 978-91-7685-585-0.

No 1854 Meysam Aghighi: Computational Complexity of some Optimization Problems in Planning , 2017, ISBN 978-91-7685-519-5.

Linköping Studies in Arts and Science No 504 Ing-Marie Jonsson: Social and Emotional

Characteristics of Speech-based In-Vehicle Information Systems: Impact on Attitude and Driving Behaviour, 2009, ISBN 978-91-7393-478-7.

No 586 Fabian Segelström: Stakeholder Engagement for Service Design: How service designers identify and communicate insights, 2013, ISBN 978-91-7519-554-4.

No 618 Johan Blomkvist: Representing Future Situations of Service: Prototyping in Service Design, 2014, ISBN 978-91-7519-343-4.

No 620 Marcus Mast: Human-Robot Interaction for Semi-Autonomous Assistive Robots, 2014, ISBN 978-91-7519-319-9.

No 677 Peter Berggren: Assessing Shared Strategic Understanding, 2016, ISBN 978-91-7685-786-1.

No 695 Mattias Forsblad: Distribu ted cognition in home environments: The prospective memory and cognitive practices of older adults, 2016, ISBN 978-91-7685-686-4.

Linköping Studies in Stat ist ics

No 9 Davood Shahsavani: Computer Experiments De-signed to Explore and Approximate Complex Deter -ministic Models, 2008, ISBN 978-91-7393-976-8.

No 10 Karl Wahlin: Roadmap for Trend Detection and As-sessment of Data Quality, 2008, ISBN 978-91-7393-792-4.

No 11 Oleg Sysoev: Monotonic regression for large multivariate datasets, 2010, ISBN 978-91-7393-412-1.

No 13 Agné Burauskaite-Harju: Characterizing Temporal Change and Inter-Site Correlations in Daily and Sub-daily Precipitation Extremes, 2011, ISBN 978-91-7393-110-6.

Linköping Studies in Informat ion Science

No 1 Karin Axelsson: Metodisk systemstrukturering- att skapa samstämmighet mellan informationssystem-arkitektur och verksamhet, 1998. ISBN-9172-19-296-8.

No 2 Stefan Cronholm: Metodverktyg och användbarhet - en stud ie av datorstödd metod baserad systemutveckling, 1998, ISBN-9172-19-299-2.

No 3 Anders Avdic: Användare och utvecklare - om anveckling med kalkylprogram, 1999. ISBN-91-7219-606-8.

No 4 Owen Eriksson: Kommunikationskvalitet hos infor-mationssystem och affärsprocesser, 2000, ISBN 91-7219-811-7.

No 5 Mikael Lind: Från system till process - kriterier för processbestämning vid verksamhetsanalys, 2001, ISBN 91-7373-067-X.

No 6 Ulf Melin: Koord ination och informationssystem i företag och nätverk, 2002, ISBN 91-7373-278-8.

No 7 Pär J. Ågerfalk: Information Systems Actability - Un-derstanding Information Technology as a Tool for Business Action and Communication, 2003, ISBN 91-7373-628-7.

No 8 Ulf Seigerroth: Att förstå och förändra system-utvecklingsverksamheter - en taxonomi för metau tveckling, 2003, ISBN91-7373-736-4.

No 9 Karin Hedström: Spår av datoriseringens värden --- Effekter av IT i äld reomsorg, 2004, ISBN 91-7373-963-

4. No 10 Ewa Braf: Knowledge Demanded for Action -

Stud ies on Knowledge Mediation in Organisations, 2004, ISBN 91-85295-47-7.

No 11 Fredrik Karlsson: Method Configuration method and computerized tool sup port, 2005, ISBN 91-85297-48-8.

No 12 Malin Nordström: Styrbar systemförvaltning - Att organisera systemförvaltningsverksamhet med hjälp av effektiva förvaltningsobjekt, 2005, ISBN 91-85297-60-7.

No 13 Stefan Holgersson: Yrke: POLIS - Yrkeskunskap, motivation, IT-system och andra förutsättningar för polisarbete, 2005, ISBN 91-85299-43-X.

No 14 Benneth Christiansson, Marie-Therese Christiansson: Mötet mellan process och komponent - mot ett ramverk för en verksamhetsnära kravspecifikation vid anskaffning av komponent-baserade informationssystem, 2006, ISBN 91-85643-22-X.

computational complexity of some optimization problems...

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