Constraint-based problem solving

Model problem specify in terms of constraints on acceptable solutions define variables (denotations) and domains define constraints in some language

Solve model define search space / choose algorithm

– incremental assignment / backtracking search

– complete assignments / stochastic search

design/choose heuristics

Verify and analyze solution

Model problem specify in terms of constraints on acceptable solutions define variables (denotations) and domains define constraints in some language

Solve model define search space / choose algorithm

– incremental assignment / backtracking search

– complete assignments / stochastic search

design/choose heuristics

Verify and analyze solution

Constraint-based problem solving

ConstraintSatisfactionProblem

Constraint satisfaction problem

A CSP is defined by a set of variables a domain of values for each variable a set of constraints between variables

A solution is an assignment of a value to each variable that

satisfies the constraints

Example: Crossword puzzles

CSP, binary CSP, SAT, 3-SAT, ILP, ...

Model and solve in one of these languages Model in one language, translate into

another to solve

Options

Article of faith

“Constraints arise naturally in most areas of human endeavor. They are the natural medium of expression for formalizing regularities that underlie the computational and physical worlds and their mathematical abstractions.”

P. Van Hentenryck and V. Saraswat

Constraint Programming: Strategic Directions

Reducibility

NP-Complete

3-SAT

ILP

CSP binary CSP

(0,1)-ILP

CSP, binary CSP, SAT, 3-SAT, ILP, ...

Model and solve in one of these languages Model in one language, translate into

another to solve

Options

Importance of the model

“In integer programming, formulating a ‘good’ model is of crucial importance to solving the model.”

G. L. Nemhauser and L. A. Wolsey

Handbook in OR & MS, 1989

“Same for constraint programming.”

Folk Wisdom, CP practitioners

Measures for comparing models

How easy is it to

–write down,

–understand,

–modify, debug,

–communicate? How computationally difficult is it to solve?

Constraint systems/languages

System LanguageILOG Solver C++ libraryCosytec CHIP C++ libraryEclipse Logic programmingOz Functional programming

Abstractions

Capture commonly occurring constraints special propagation algorithms e.g. alldifferent,

cardinality,

cumulative, ... User-defined constraints

full power of host language

CSP versus ILP

OO databases versus Relational databases C++ versus C C++ versus Fortran C++ versus Assembly language

Computational difficulty?

What is a good model depends on algorithm Choice of variables defines search space Choice of constraints defines

how search space can be reduced how search can be guided

Improving model efficiency

Given a model:

Add/remove variables, values, constraints (keep the denotation of the variables)

Use/translate to a different representation(change the denotation of the variables)

Adding redundant constraints

Improve computational efficiency of model by adding “right” constraints symmetries removed dead-ends encountered earlier in search process

Three methods: add hand-crafted constraints during modeling apply a consistency algorithm before solving learn constraints while solving

Examples:

Adding hand-crafted constraints: crossword puzzles

Applying a consistency algorithm: dual representation

Adding redundant variables

Variables that are abstractions of other variables e.g, decision variables

Suppose x has domain {1,…,10}

Add Boolean variable to represent decisions

(x < 5), (x 5) Variables that represent constraints

Translations between models

Improve computational efficiency by completely changing the model change denotation of variables

– e.g, convert from non-binary to binary

aggregate variables– e.g, timetabling multiple sections of a course

Not much as has been done on either the theory or the practice side

Conversion to binary

Any CSP can be converted into one with only binary constraints

Two techniques known dual graph method (Dechter & Pearl, 1989)

hidden variable method (Peirce, 1933; Dechter 1990)

Translations are polynomial if constraints are represented extensionally

(x1 x2 x6) (x1 x3 x4) (x4 x5 x6) (x2 x5 x6) Non-binary CSP: Boolean variables: x1, …, x6

constraints: one for each clause

3-SAT example

C1(x1, x2, x6) = {(0,0,1), (0,1,0), (0,1,1),

(1,0,0), (1,0,1), (1,1,0), (1,1,1)}

C2(x1, x3, x4) = ...

Dual graph representation

y1 = (x1, x2, x6) y4 = (x2, x5, x6)

y2 = (x1, x3, x4) y3 = (x4, x5, x6)

x2, x6

x5, x6

x4

x6x1

Hidden variable representation

x1 x2 x3 x4 x5 x6

h1

h2 h3

h4

Non-binary constraint:

Augmented constraint:

Binary constraints:

Hidden variable representation

C1(x1, x2, x6) = {(0,0,1), (0,1,0), (0,1,1),

(1,0,0), (1,0,1), (1,1,0),

(1,1,1)}

C1(x1, x2, x6, h1) = {(0,0,1,0), (0,1,0,1), (0,1,1,2),

(1,0,0,3), (1,0,1,4), (1,1,0,5), (1,1,1,6)}

R1(x1, h1) = {(0,0), (0,1), (0,2), (1,3), (1,4), (1,5), (1,6)}

R2(x2, h1) = {(0,0), (1,1), (1,2), (0,3), (0,4), (1,5), (1,6)}

R3(x6, h1) = {(1,0), (0,1), (1,2), (0,3), (1,4), (0,5), (1,6)}

x1

x2

x3

h1

h2

h3

Dual exponentially worse

FC on non-binary:

O(n) consistency checks

FC on dual:

O(2n) consistency

checks

x1, x1 x2, x1 x2 x3, ..., x1 … xn-1 xn

x1

x2

x3

h1

h2

h3

Dual exponentially better

FC on non-binary:

O(n2n) consistency checks

FC on dual:

O(n2) consistency

checks

x1 … xn-1, x1 … xn-2 xn, …, x2 … xn

Worst case:

Best case:

Bounds on performance

FC(non-binary) FC+(hidden) dk

FC+(hidden) FC (non-binary) dn

Crossword puzzles

1 2 3

6

4

7

5

8

10

9

20

11

22

12

21

13

17

14

181615

23

19

aaardvarkabackabacusabaftabaloneabandon...

Mona Lisamonarchmonarchymonarda...zymurgyzyrianzythum

Crossword puzzles

puzzle orig. cc’s dual cc’s n m

1 52 292 4 3

2 138 3,267 6 8

3 185 11,404 8 14

4 208 10,481 12 14

5 12,509 18,686 10 19

6 12,636 19,876 14 21

7 556,660 28,962 14 23

Crossword puzzles

puzzle orig. cc’s dual cc’s n m

8 5,780,710 27,656 20 30

9 * 47,167 26 41

10 * 224,258 18 35

11 * 29,777 24 40

12 * 729,125 18 38

13 28,446,460 50,281 28 52

14 * 72,307 34 65

Model of random non-binary CSP

n variables each with domain size d

m constraints each with k variables, chosen at random each with t tuples, chosen at random

Order of magnitude curves(n

um

ber

of

const

rain

ts)

/ (n

um

ber

of

vari

able

s)

(number of tuples in constraints) / (maximum tuples)

n = 20d = 2k = 3

n = 20d = 10k = 3

n = 20d = 10k = 5

n = 20d = 2k = 5

Random 3-SAT

clauses/variables

costratio

0.25 0.9

0.5 1.5

1 2.5

2 4.2

4 31

8 33

16 38

Given start state, goal state, and actions: determine a plan (a sequence of actions)

Box1

Project: Planning

Box2Box2

Contrasts

CP philosophy/methodology emphasis on modeling, domain knowledge general purpose search algorithm

– backtracking with constraint propagation

Successful e.g., can solve practical scheduling problems

Contrasts

Planning philosophy/methodology emphasis on minimal model

– just representation of actions

special-purpose search algorithms Not as successful

has not solved many practical problems

Tradeoffs

Robust CSP model needed for each new domain can require much intellectual effort

Less work needs to be done on algorithms many general purpose constraint solvers available

CSP model of planning

State-based model model each state by a collection of variables constraints enforce valid transitions between states

Example: logistics world variable for each package, truck, plane domains of packages: all locations, trucks, planes domains of trucks, planes: all locations

CSP model: constraints (I)

Action constraints model the effects of actions patterned after explanation closure axioms

State constraints variables within a state must be consistent

Improving model efficiency

Can add/remove/aggregate/decompose variables domain values constraints

Here: added hidden variables added redundant & symmetry-breaking constraints

CSP model: constraints (II)

Symmetric values constraints break symmetries on values variables can be

assigned Action choice constraints

break symmetries on equivalent permutations of actions

CSP model: constraints (III)

Domain constraints restrictions on original domains of variables

Capacity constraints bounds on resources

Distance constraints bounds on steps needed for a variable to change

from one value to another

Solving the CSP model

Given an instance of a planning problem generate a model with one step in it instantiate variables in the initial and goal states search for a solution (GAC+CBJ) repeat, incrementing number of steps, until plan

is found Properties:

forwards, backwards, or middle out planner sound, complete, guaranteed to terminate

Experiments

Five test domains from AIPS’98 Five planners:

CPlan Blackbox (Kautz & Selman) HSP (Bonet & Geffner) IPP (Koehler & Nebel) TLPlan (Bacchus & Kabanza)

Setup: machines: 400MHz Pentium II’s resources: 1 hour CPU time, 256 Mb memory

Gripper problems

CPlan BBox HSP IPP

1 0.0 0.1 0.0 0.0

2 0.0 5.7 0.1 0.4

3 0.1 0.1 7.9

4 0.2 0.2 100.4

5 0.3 0.3

6 0.5 0.4

20/20 2/20 20/20 4/20

Logistics problems

CPlan BBox HSP IPP

1 0.1 1.5 0.6

2 0.1 4.3 552.2

3 0.9

4 0.2

5 1.7 148.0 1.2 2.5

6 18.9

28/35 8/35 5/35 10/35

Mystery problems

CPlan BBox HSP IPP

1 0.0 0.1 0.1 0.1

2 0.0 4.2 7.1 11.4

3 0.0 0.4 0.4 0.9

4 0.0 1.2 0.4

5 0.0 11.4 7.6

6 3.1 134.0

30/30 19/30 16/30 24/30

MysteryPrime problems

CPlan BBox HSP IPP

1 0.1 0.6 0.1 2.3

2 0.1 4.4 13.9 23.9

3 0.2 0.6 0.5 4.4

4 0.2 0.6 0.6 2.5

5 0.2 7.2

6 2.3 201.8

35/35 18/35 25/35 21/35

Grid problems

CPlan BBox HSP IPP

1 0.7 8.1 1.1 3.0

2 33.4 4.3 9.4

3

4 1773.3 57.0

5

Related work

Planning as a CSP satisfiability (Kautz & Selman)

ILP (e.g., Bockmayr & Dimopoulos)

Adding declarative knowledge to improve efficiency hand-coded (e.g, Kautz & Selman, Bacchus &

Kabanza) automatically derived (e.g., Fox & Long, Nebel et al.)

Project: Resource-constrained instruction scheduling (I)

Modern architectures (VLIW) allow instruction level parallelism multiple functional units

Compiler to generate code that takes advantage of parallelism resource-constrained scheduling task

a: j i + hb: k i + gc: l j + 1

1: a, b2: c

Project: Resource-constrained instruction scheduling (II)

Current methods: heuristic recent: integer linear programming approach

Project: investigate constraint programming approach GNU compiler, MERCED chip

Project: Theoretical analysis of alternative models

Improve computational efficiency by completely changing the model change denotation of variables

– e.g, convert from non-binary to binary

aggregate variables– e.g, timetabling multiple sections of a course

Not much as has been done on either the theory or the practice side

Conclusions

Importance of the model ease of modeling is important form of model is important

Constraint programming advantages succinctness, declarativeness of models flexibility in specifying model on some classes of problems: speed

Conclusions

Advantages shared with CSP-like approaches expressiveness of modeling language declarativeness of models independence of model and solving algorithm

Advantages over other CSP-like approaches succinctness of models robustness: scales well, not as brittle speed

Future Work

Approximate planning heuristic CSP models solve same CSP models using local search

Alternative CSP models action-based models vs state-based models finding the right model can be key to solving

difficult combinatorial problems