computational complexity for social choice theorists jörg rothe comsoc 2008, liverpool, uk

Computational Complexity Computational Complexity for Social Choice Theoristsfor Social Choice Theorists

Jörg RotheJörg Rothe

COMSOC 2008, Liverpool, UK COMSOC 2008, Liverpool, UK

Question:Question: What do you do in complexity theory?What do you do in complexity theory?

Answers:Answers:

•Struggling with Struggling with intractable problemsintractable problems..

Everything you Always Wanted to Know about Everything you Always Wanted to Know about Complexity Theory but Were Afraid to AskComplexity Theory but Were Afraid to Ask



Answers:Answers:


•Collecting them in Collecting them in complexity classescomplexity classes and making up funny names for those.and making up funny names for those.

SHEEPSHEEPCATSCATS

CHICKENSCHICKENS

CATTLECATTLE

DOGDOGSS

Scott Aaronson‘s Zoo Scott Aaronson‘s Zoo of Complexity Classesof Complexity Classes



Answers:Answers:



•Comparing the complexity of problems Comparing the complexity of problems via via reducibilitiesreducibilities to find the „hardest“ to find the „hardest“ problems in the class: problems in the class: CompletenessCompleteness..

PPmm

PPmm

PPmm

•Polynomial-time Many-One Reducibility: A B (fFP)(xΣ*)[xA f(x)B].•B is B is hard for class hard for class CC if if (ACC)[A B].•B is C-C-complete if B is in C C and is hard for and is hard for C C ..

PPmm

PPmm


Question:Question: What else do you do in complexity theory?What else do you do in complexity theory?

Answers:Answers:



•Comparing the complexity of problems Comparing the complexity of problems via via reducibilitiesreducibilities to find the „hardest“ to find the „hardest“ problems in the class: problems in the class: CompletenessCompleteness..

•Studying Studying hierarchies of complexity hierarchies of complexity classesclasses, such as, such as

•the the Polynomial HierarchyPolynomial Hierarchy,,

•the the Boolean Hierarchy over NPBoolean Hierarchy over NP, etc., etc.

„Computer Science is not about computers,any more than astronomy is about telescopes.“

Edsger Dijkstra

OutlineOutline

• Everything You Always Wanted to Know about...Everything You Always Wanted to Know about...• Some Problems from Social Choice TheorySome Problems from Social Choice Theory

• Voting Problems: Winner Determination, Manipulation, Control, ...Voting Problems: Winner Determination, Manipulation, Control, ...• Power-Index Comparison and Weighted Voting GamesPower-Index Comparison and Weighted Voting Games• Multiagent Resource AllocationMultiagent Resource Allocation

• Foundations of Complexity TheoryFoundations of Complexity Theory• Problems Complete for NPProblems Complete for NP• Parallel Access to NP and the Polynomial HierarchyParallel Access to NP and the Polynomial Hierarchy• Probabilistic Polynomial Time and Power IndicesProbabilistic Polynomial Time and Power Indices• DP and the Boolean Hierarchy over NPDP and the Boolean Hierarchy over NP

Make the List ... by the Plurality Rule:Make the List ... by the Plurality Rule:Rank 1: Rank 1: JJRank 2: Rank 2: D D and and KK (aequo loco) (aequo loco)Rank 3: C and H (aequo loco)Rank 3: C and H (aequo loco)

Voting Problems: Voting Problems: How to Recruit a new Faculty MemberHow to Recruit a new Faculty Member

Preferences of the Recruiting Committee:Preferences of the Recruiting Committee:

JJ < A < B < E < < A < B < E < DD < F < G < H < < F < G < H < KK < I < C < I < C

I < I < JJ < A < < A < DD < E < F < G < B < C < < E < F < G < B < C < K K < H < H

A < B < F < G < H < A < B < F < G < H < KK < I < C < < I < C < JJ < E < < E < DD

E < G < F < B < E < G < F < B < JJ < I < H < C < A < < I < H < C < A < DD < < KK

C < A < F < E < B < C < A < F < E < B < KK < H < G < I < < H < G < I < DD < < JJ


H < G < H < G < KK < I < C < B < A < F < < I < C < B < A < F < JJ < E < < E < DD

DD < I < E < A < B < H < F < G < C < < I < E < A < B < H < F < G < C < JJ < < KK

F < G < F < G < DD < I < E < B < H < A < C < < I < E < B < H < A < C < KK < < JJ

Candidates:Candidates: A, B, C, A, B, C, DD, E, F, G, H, I, , E, F, G, H, I, JJ, , KK

Make the List ... by Borda‘s Rule:Make the List ... by Borda‘s Rule:Rank 1: Rank 1: KK (63 points) (63 points)Rank 2: Rank 2: J J (60 points)(60 points) Rank 3: Rank 3: D D (56 points)(56 points)

Make the List ... by the Majority Rule:Make the List ... by the Majority Rule:Rank 1: Rank 1: DD and and JJ and and KK (aequo loco) (aequo loco)

Since: Since: DD defeats defeats JJ by 5:4 votes, by 5:4 votes, JJ defeats defeats KK by 5:4 votes, by 5:4 votes, KK defeats defeats DD by 5:4 votes. by 5:4 votes. Condorcet‘s ParadoxonCondorcet‘s Paradoxon

D D CycleCycle

K K JJ

Voting Problems: Voting Problems: Winner Determination, Manipulation, Control, BriberyWinner Determination, Manipulation, Control, Bribery

Winner DeterminationWinner Determination::•How hard is it to determine the winners of a given election?How hard is it to determine the winners of a given election?•For most election systems, it is easy to determine the winners,For most election systems, it is easy to determine the winners, but for some it is hard (Carroll, Kemeny, and Young elections).but for some it is hard (Carroll, Kemeny, and Young elections).

ManipulationManipulation::•How hard is it, computationally, to manipulate the result ofHow hard is it, computationally, to manipulate the result of an election by strategic voting?an election by strategic voting?•The Gibbard-Satterthwaite Theorem says: Manipulation isThe Gibbard-Satterthwaite Theorem says: Manipulation is unavoidable in principle.unavoidable in principle.

ControlControl::•How hard is it, computationally, for an evil chair to influenceHow hard is it, computationally, for an evil chair to influence the outcome of an election via procedural changes?the outcome of an election via procedural changes?

BriberyBribery::•How hard is it, computationally, for an external agent to bribeHow hard is it, computationally, for an external agent to bribe certain voters in order to change an election‘s outcome?certain voters in order to change an election‘s outcome?

Voting Problems: Voting Problems: Winner Determination, Manipulation, Control, BriberyWinner Determination, Manipulation, Control, Bribery

Winner DeterminationWinner Determination: : Hardness is undesirable!Hardness is undesirable!•How hard is it to determine the winners of a given election?How hard is it to determine the winners of a given election?•For most election systems, it is easy to determine the winners,For most election systems, it is easy to determine the winners, but for some it is hard (Carroll, Kemeny, and Young elections).but for some it is hard (Carroll, Kemeny, and Young elections).

ManipulationManipulation: : Hardness provides protection!Hardness provides protection!•How hard is it, computationally, to manipulate the result ofHow hard is it, computationally, to manipulate the result of an election by strategic voting?an election by strategic voting?•The Gibbard-Satterthwaite Theorem says: Manipulation isThe Gibbard-Satterthwaite Theorem says: Manipulation is unavoidable in principle.unavoidable in principle.

ControlControl: : Hardness provides protection!Hardness provides protection!•How hard is it, computationally, for an evil chair to influenceHow hard is it, computationally, for an evil chair to influence the outcome of an election via procedural changes?the outcome of an election via procedural changes?

BriberyBribery: : Hardness provides protection!Hardness provides protection!•How hard is it, computationally, for an external agent to bribeHow hard is it, computationally, for an external agent to bribe certain voters in order to change an election‘s outcome?certain voters in order to change an election‘s outcome?

Please attend thePlease attend theafternoon session tomorrowafternoon session tomorrow

to learn more aboutto learn more aboutbribery and control.bribery and control.

Power-Index Comparison and Power-Index Comparison and Weighted Voting GamesWeighted Voting Games

20 papers

20 papers

50 papers $2M $5M $2M

4 papers$10M

Harvard University Money University

Where will I have more

(local) power?

Aha! Clearly, I will have more (local) power at

Money University! But how else can I justify this choice?

Power-Index Comparison and Power-Index Comparison and Weighted Voting GamesWeighted Voting Games

Power Index idea: How “often” is the given player critical to the winning side?

Alice

3

Bob

3

Carol

4

Alice

3

Bob

3

Carol

4

Alice

3

Bob

3

Carol

6

Alice

2

Bob

2

Equal power No power Total power

Power indices (e.g., Power indices (e.g., Shapley-ShubikShapley-Shubik and and BanzhafBanzhaf) formally ) formally capture this idea. capture this idea. How hard is it toHow hard is it to

•compute a power index for a given weighted voting game?compute a power index for a given weighted voting game?

•compare the power index of two given weighted voting games?compare the power index of two given weighted voting games?

Weighted Voting Games:

Multiagent Resource AllocationMultiagent Resource Allocationafter World War IIafter World War II

• Set of Agents: the Allies of World War II

• Set of Resources: Germany‘s Federal States

Multiagent Resource AllocationMultiagent Resource Allocation

• Set of Agents: A = {1, 2, ..., n}• Set of Resources: R = { }mrrr ,...,, 21

• Each agent a has• a preference over allocations• a utility function that assigns values to bundles of resources.

• Each resource is indivisible and nonsharable.• An allocation is a mapping P from A to bundles of resources.

Useful properties:• Envy-freeness• Pareto optimality

Given agents Given agents AA, resources , resources RR, and utility , and utility functions functions UU, how hard is it to, how hard is it to

•to maximize (utilitarian) social welfare?to maximize (utilitarian) social welfare?

•to determine if a given allocation is to determine if a given allocation is Pareto-optimal?Pareto-optimal?

•to determine if a given allocation is envy-to determine if a given allocation is envy-free?free?

a


Edsger Dijkstra

OutlineOutline




Foundations of Complexity TheoryFoundations of Complexity Theory

1912 - 1954

Alan TuringAlan Turing• Broke the Broke the

Enigma-CodeEnigma-Code• Invented the Invented the

Turing machineTuring machine

A problem‘s computational complexity is determined by:•computational model

•Turing machine•Boolean circuit•...

•computational paradigm •Deterministic TM•Nondeterministic TM•Probabilistic TM•Alternating TM•...

•complexity measure(a.k.a. resource) used

•computation time•space (memory)•... (see Blum‘s axioms)

How to get a problem into the computer?How to get a problem into the computer?

What is a Turing machine?What is a Turing machine?

Which problems are not solvable on a Which problems are not solvable on a computer?computer?• The (deterministic, worst-case) complexity measureThe (deterministic, worst-case) complexity measure TimeTime of a of a

Turing machine Turing machine M M gives, as a function of the input size gives, as a function of the input size nn, the , the maximum number of steps maximum number of steps M M needs on inputs of sizeneeds on inputs of size n. n.

• The (deterministic, worst-case) complexity measure The (deterministic, worst-case) complexity measure SpaceSpace of a of a Turing machine Turing machine M M gives, as a function of the input size gives, as a function of the input size nn, the , the maximum number of tape cells maximum number of tape cells M M needs on inputs of sizeneeds on inputs of size n. n.

Compare with Compare with Impossibility TheoremsImpossibility Theorems from Social from Social Choice:Choice:•Arrow:Arrow: No election system satisfying a certain small set ofNo election system satisfying a certain small set of „ „fairness“ conditions can be nondictatorial.fairness“ conditions can be nondictatorial.•Gibbard-Satterthwaite:Gibbard-Satterthwaite: Manipulation is unavoidable Manipulation is unavoidable in principle.in principle.

Turing machines:Turing machines:

•capture everything computableeverything computable

•are a simple, abstract model of a model of a computer/algorithmcomputer/algorithm

•form the theoretical basistheoretical basis of computer science

•facilitate the complexity analysiscomplexity analysis


Edsger Dijkstra

OutlineOutline




• Complexity classes Complexity classes collect all problems solvable on a Turing collect all problems solvable on a Turing machine of a certain type within a certain amount of resourcesmachine of a certain type within a certain amount of resources

• PP is the class of polynomial-time („efficiently“) solvable problems

• NPNP is the class of problems with efficiently checkable solutions

NNondeterministicondeterministicPPolynomial Timeolynomial Time

x L

A R A

• Central open question in computer science:

P = NP ?P = NP ?

• One of the standard NP-complete NP-complete problems: Traveling Sales Traveling Sales PersonPerson • TSP TSP belongs to NP NP (upper bound)(upper bound)• TSP TSP is one of the „hardest“ problems in NPNP, i.e., every

problem in NPNP efficiently reduces to TSP TSP (lower bound)(lower bound)

LondonLondon

The Traveling Salesperson ProblemThe Traveling Salesperson Problem

BerlinBerlinDüsseldorfDüsseldorf

ParisParis

508

575538

919

338 871

Tour 1: D-B-L-P-D = 2340Tour 1: D-B-L-P-D = 2340

Tour 2: D-P-B-L-D = 2836Tour 2: D-P-B-L-D = 2836

Tour 3: D-L-P-B-D = 2322 Tour 3: D-L-P-B-D = 2322 is optimal.is optimal.

• For n cities:

((n n - 1)! - 1)! // 2 tours 2 tours• For n = 1000: 22xx10 tours10 tours• There are about 10 atoms in the 10 atoms in the

universeuniverse

7777

25642564

Voting Problems: ManipulationVoting Problems: Manipulation

Preference profile:Preference profile: Multiset of voters‘ preferencesMultiset of voters‘ preferences

JJ < A < B < E < < A < B < E < DD < F < G < H < < F < G < H < KK < I < C < I < C

I < I < JJ < A < < A < DD < E < F < G < B < C < < E < F < G < B < C < K K < H < H

A < B < F < G < H < A < B < F < G < H < KK < I < C < < I < C < JJ < E < < E < DD

E < G < F < B < E < G < F < B < JJ < I < H < C < A < < I < H < C < A < DD < < KK



H < G < H < G < KK < I < C < B < A < F < < I < C < B < A < F < JJ < E < < E < DD

DD < I < E < A < B < H < F < G < C < < I < E < A < B < H < F < G < C < JJ < < KK

F < G < F < G < DD < I < E < B < H < A < C < < I < E < B < H < A < C < KK < < JJ

Candidates:Candidates: A, B, C, A, B, C, DD, E, F, G, H, I, , E, F, G, H, I, JJ, , KK

Preference relation:Preference relation:

• strict,strict,

• transitive,transitive,

• completecomplete..

Manipulation: Manipulation: Strategic voters misrepresent their Strategic voters misrepresent their preferences to change the election‘s outcome, either topreferences to change the election‘s outcome, either to•make their favorite candidate win (make their favorite candidate win (constructive caseconstructive case) or to) or to•prevent a despised candidate‘s victory (prevent a despised candidate‘s victory (destructive casedestructive case). ).

Election Systems that are NP-hard to Election Systems that are NP-hard to ManipulateManipulate

Manipulation ProblemManipulation ProblemInstance: (C,c,V), where C is a set of candidates, V is the voters‘ preference profile over C, c a designated candidate in C.Question: Does there exist a preference order making c a winner?

Gibbard-Satterthwaite:Gibbard-Satterthwaite: Manipulation is unavoidable Manipulation is unavoidable in principle.in principle.

J. Bartholdi, C. Tovey & M. Trick (SCW 1989):J. Bartholdi, C. Tovey & M. Trick (SCW 1989):

ForFor Second-Order Copeland, Second-Order Copeland, the winner problem is efficientlythe winner problem is efficiently

solvable, but the manipulation problem is NP-complete.solvable, but the manipulation problem is NP-complete.

V. Conitzer, T. Sandholm & J. Lang (J.ACM 2007):V. Conitzer, T. Sandholm & J. Lang (J.ACM 2007):• Studied Studied coalitionalcoalitional manipulation by manipulation by weightedweighted voters voters• Characterized the exact number of candidates for which Characterized the exact number of candidates for which

manipulation becomes NP-hard for plurality, Borda, STV, manipulation becomes NP-hard for plurality, Borda, STV, Copeland, maximin, veto, and other protocolsCopeland, maximin, veto, and other protocols

• Considered both Considered both constructiveconstructive and and destructivedestructive manipulation manipulation

Election Systems that are NP-hard to Election Systems that are NP-hard to ManipulateManipulate

E. Hemaspaandra & L. Hemaspaandra (JCSS 2007):E. Hemaspaandra & L. Hemaspaandra (JCSS 2007):Provided the first dichotomy result for voting systems:an easy-to-check condition („diversity of dislike“) that separates

• Scoring protocols that are NP-hard to manipulate from• Scoring protocols that are easy to manipulate.

C. Dwork, R. Kumar, M. Naor & D. Sivakumar (WWW 2001):C. Dwork, R. Kumar, M. Naor & D. Sivakumar (WWW 2001):

„„Rank Aggregation Methods for the Web“:Rank Aggregation Methods for the Web“:• Kemeny SCFKemeny SCF is suitable to preventis suitable to prevent „manipulation „manipulation of website of website

rankings“rankings“ by search engines. by search engines.• Efficient heuristic: „Local Kemenization“.Efficient heuristic: „Local Kemenization“.

Holy GrailHoly Grail

P. Faliszewski, E. Hemaspaandra & H. Schnoor (AAMAS 2008):P. Faliszewski, E. Hemaspaandra & H. Schnoor (AAMAS 2008):Established NP-hardness results for coalitional manipulation both

forweighted and unweighted voters within (various) Copeland

elections.


Edsger Dijkstra

OutlineOutline




The Condorcet PrincipleThe Condorcet Principle

• Majority RuleMajority Rule::

Candidate A defeats candidate B if A gets more votes than B.Candidate A defeats candidate B if A gets more votes than B.• A A Condorcet candidateCondorcet candidate defeats every other candidate defeats every other candidate

according to the majority rule.according to the majority rule.

Example 1Example 1

Voter 1: A < B < Voter 1: A < B < CC

Voter 2: A < Voter 2: A < CC < B < B

Voter 3: B < Voter 3: B < CC < A < A

CC defeats A and B by 2:1 and defeats A and B by 2:1 and thus is a thus is a Condorcet candidate.Condorcet candidate.

Example 2Example 2

Voter 1: A < B < CVoter 1: A < B < C

Voter 2: C < A < BVoter 2: C < A < B

Voter 3: B < C < AVoter 3: B < C < A

There‘s NOThere‘s NO Condorcet winner!Condorcet winner!

Condorcet‘s ParadoxCondorcet‘s Paradox

Condorcet PrincipleCondorcet Principle

An election system should respect the notion of Condorcet winner.An election system should respect the notion of Condorcet winner.

AA CycleCycle

BB CC

Condorcet SCFs...Condorcet SCFs...

... respect the Condorcet Principle by choosing the ... respect the Condorcet Principle by choosing the Condorcet Condorcet CandidateCandidate whenever one exists. whenever one exists.

Lewis Carroll‘s Voting System (1876):Lewis Carroll‘s Voting System (1876): The winnerThe winner is whoever becomes a Condorcet candidate

by a minimumminimum number of sequential switches of adjacent candidates in the voters‘ preference profile.

H. P. Young‘s Voting System (1977):H. P. Young‘s Voting System (1977): The winnerThe winner is whoever becomes a Condorcet candidate

by removing a minimumminimum number of voters from the preference profile.

J. G. Kemeny‘s Voting System (1959):J. G. Kemeny‘s Voting System (1959): The winnerThe winner is the candidate ranked first place in the

„Consensus Ranking,“ a preference order that minimizesminimizes the sum of the distances to the voters‘ preferences in the profile.

......

Carroll ElectionsCarroll Elections

• The The Carroll scoreCarroll score of a candidateof a candidate C C is the smallest number of is the smallest number of

sequential switches of adjacent candidates in the preferencesequential switches of adjacent candidates in the preference

profile of the voters that make profile of the voters that make CC a Condorcet candidate. a Condorcet candidate.

• Carroll winnerCarroll winner is whoever has the lowest Carroll score.is whoever has the lowest Carroll score.

Example: Example: Carroll scoreCarroll score

Voter 1: A < Voter 1: A < BB < < CC

Voter 2: A < Voter 2: A < CC < < BB

Voter 3: Voter 3: CC < A < < A < BB

Voter 4: Voter 4: CC < < BB < A < A

BB defeats A and defeats A and CC by 3:1 and by 3:1 and so is a so is a Condorcet candidateCondorcet candidate

Score(Score(BB) = 0) = 0Example: Example: Carroll scoreCarroll score


Voter 2: A < Voter 2: A < B B < < CC


Voter 4: Voter 4: CC < < BB < A < A

CC ties A and ties A and BB (2:2) and thus is (2:2) and thus is no no Condorcet candidateCondorcet candidate





Voter 4: Voter 4: BB < < CC < A < A

CC defeats defeats BB (3:1), ties A (2:2): (3:1), ties A (2:2): No No Condorcet candidateCondorcet candidate





Voter 4: Voter 4: BB < A < < A < CC

CC defeats A and defeats A and BB by 3:1 and by 3:1 and so is a so is a Condorcet candidateCondorcet candidate

Score(Score(CC) = 3) = 3

Score(A) = 3Score(A) = 3

For this preference profile For this preference profile P, P, thethe Carroll SCF gives: Carroll SCF gives:

A = A = C C < < BB

Problems for Carroll ElectionsProblems for Carroll Elections

Carroll WinnerCarroll WinnerInstance: A Carroll triple (C,c,V), where C Set of Candidates, V Preference profile of voters over C, c a designated candidate in C.Question:

? )Score()Score(

all for that true it is is, That winner?Carroll a Is

dc

C, dc

Carroll RankingCarroll RankingInstance: A Carroll triple (C,c,V) and another candidate d in C.Question: ? )Score()Score( that true it Is dc

Carroll ScoreCarroll ScoreInstance: A Carroll triple (C,c,V) and a positive integer k.Question: ? )Score( that true it Is kc

Question: Question:

Can we do better?Can we do better?

Results for Carroll Election ProblemsResults for Carroll Election ProblemsJ. Bartholdi, C. Tovey & M. Trick (SCW 1989):J. Bartholdi, C. Tovey & M. Trick (SCW 1989):

• Carroll Score and Kemeny Score are NP-complete.

• Carroll Winner and Kemeny WinnerCarroll Winner and Kemeny Winner are NP-hard.

E. Hemaspaandra, L. Hemaspaandra & J. Rothe (J.ACM 1997):E. Hemaspaandra, L. Hemaspaandra & J. Rothe (J.ACM 1997):

• Carroll WinnerCarroll Winner and Carroll Ranking Carroll Ranking are complete for

P P :: „parallel access to NP.“„parallel access to NP.“||||

NPNP

J. Rothe, H. Spakowski & J. Vogel (TOCS 2003):J. Rothe, H. Spakowski & J. Vogel (TOCS 2003):

• Young WinnerYoung Winner and Young Ranking Young Ranking are PP -complete.||||

NPNP

E. Hemaspaandra, H. Spakowski & J. Vogel (TCS 2005):E. Hemaspaandra, H. Spakowski & J. Vogel (TCS 2005):

• Kemeny WinnerKemeny Winner and Kemeny Ranking Kemeny Ranking are PP -complete.NPNP

||||

The Polynomial HierarchyThe Polynomial Hierarchy

Defining the Polynomial Hierarchy:Defining the Polynomial Hierarchy:

• Level 0: Level 0: PP (deterministic polynomial time)(deterministic polynomial time)• Level 1 has two classes:Level 1 has two classes:

• NPNP (nondeterministic polynomial time)(nondeterministic polynomial time)• coNP coNP (the class of complements of problems in NP)(the class of complements of problems in NP)

• Level 2 has two classes:Level 2 has two classes:• (nondeterministic polynomial time)(nondeterministic polynomial time)• (the class of complements of problems in )(the class of complements of problems in )coNPcoNPNPNP

NPNPNNPP NPNP NPNP

• Level k has two classes:Level k has two classes:• NP with a stack of k-1 NP oracle computationsNP with a stack of k-1 NP oracle computations• coNP with a stack of k-1 NP oracle computationscoNP with a stack of k-1 NP oracle computations

• PH PH is the union of all these levels.is the union of all these levels.

The levels of the PH capture the idea of a bounded number The levels of the PH capture the idea of a bounded number of alternating polynomially length-bounded and of alternating polynomially length-bounded and

quantors.quantors.

The Polynomial HierarchyThe Polynomial Hierarchy

Parallel and Sequential Access to NPParallel and Sequential Access to NP

NP oracle

1q 2q 3qx

321 ,, qqq 0,1,0Query listQuery list Answer listAnswer list

AcceptAccept

Parallel Access to an NP oracle:Parallel Access to an NP oracle:

Sequential Access to an NP oracle:Sequential Access to an NP oracle:•Queries may depend on answers to previousQueries may depend on answers to previous queries, which results in a queries, which results in a query treequery tree•More powerful classMore powerful class

||||NPNP

PP

NPNPPP

is the closure of NP under pol-timetruth-table reductions

is the closure of NP underpol-time Turing reductions

Proof Sketch for Carroll Winner:Proof Sketch for Carroll Winner:

Wagner‘s ToolWagner‘s Tool

Lemma 1 (Wagner‘s Tool for Lemma 1 (Wagner‘s Tool for PP -hardness-hardness))||||NPNP

Let A be some NP-complete problem, and let B be any set. If there is a polynomial-time computable function such that, for all k and all strings satisfying that

we have is odd

then B is P -hard.

kxxx 221 ,...,,AxAx jj 1 )21( kj

NPNP||||

Axi i: Bxxxf k ),...,,( 221

f

E. Hemaspaandra, L. Hemaspaandra & J. Rothe (J.ACM 1997):E. Hemaspaandra, L. Hemaspaandra & J. Rothe (J.ACM 1997):

• Carroll WinnerCarroll Winner and Carroll Ranking Carroll Ranking are complete for

P P :: „parallel access to NP.“„parallel access to NP.“||||

NPNP

Proof Sketch for Carroll Winner:Proof Sketch for Carroll Winner:Controlled Reduction and Summing Controlled Reduction and Summing

ElectionsElections Lemma 2 (Controlled Reduction to Lemma 2 (Controlled Reduction to Carroll ScoreCarroll Score))

There is a polynomial-time computable function that reduces the NP-complete problem 3-Dimensional Matching to Carroll Score such that, for all ,• has an odd number of voters.•If 3-Dimensional Matching then .•If 3-Dimensional Matching then .

),,,()( kVcCxg

g

*x

x kxScore )(x 1)( kxScore

Lemma 3 (Summation of Lemma 3 (Summation of Carroll ScoresCarroll Scores))

There is a polynomial-time computable function such that for all k and all Carroll triples each having an odd number of voters, it holds that• is a Carroll triple with an odd number of voters, and

• .

),,()),,(),...,,,(( 111 VcCVcCVcCSum kkk

),,(),,( jjj

j

VcCScoreVcCScore

Sum),,(),...,,,( 111 kkk VcCVcC

Proof Sketch for Carroll Winner:Proof Sketch for Carroll Winner:Two-Election Ranking and Merging Two-Election Ranking and Merging

ElectionsElections Lemma 4 (Two-Election Ranking)Lemma 4 (Two-Election Ranking)

The problem Two-Election Ranking:

is complete for parallel access to NP.

Instance: A pair of Carroll triples, and , with and each having an odd number of voters.

Question: Is it true that ?

),,( VcC ),,( WdD

),,(),,( WdDScoreVcCScore dc

Lemma 5 (Merging Elections)Lemma 5 (Merging Elections)There is a polynomial-time computable function such that for all Carroll triples and with and each having an odd number of voters, it holds that• is a Carroll triple,• ,• , and• for each .

Merge),,( VcC ),,( WdD dc

),,()),,(),,,(( UcBWdDVcCMerge 1),,(),,( VcCScoreUcBScore

1),,(),,( WdDScoreUdBScore),,(),,( UcBScoreUeBScore dcBe ,

Example of one Construction: Merging Example of one Construction: Merging ElectionsElections

Proof Sketch for Carroll Winner:Proof Sketch for Carroll Winner:

OverviewOverview

E. Hemaspaandra, L. Hemaspaandra & J. Rothe (J.ACM E. Hemaspaandra, L. Hemaspaandra & J. Rothe (J.ACM 1997):1997):

Carroll WinnerCarroll Winner is complete for P P :: „parallel access to „parallel access to NP.“NP.“

||||NPNP

Easy upper bound argument

Low

er b

ou

nd

arg

um

en

t

Via Lemma 1 (Wagner‘s Lemma 1 (Wagner‘s

Tool for Tool for PP -hardness-hardness))||||NPNP

Lemma 2 (Controlled Lemma 2 (Controlled Reduction to Reduction to Carroll ScoreCarroll Score))

Lemma 3 (SummationLemma 3 (Summationof of Carroll ScoresCarroll Scores))

Lemma 4Lemma 4(Two-Election Ranking)(Two-Election Ranking)

Lemma 5Lemma 5(Merging Elections)(Merging Elections)

Homogeneous Voting SystemsHomogeneous Voting Systems

P. Fishburn showed that:P. Fishburn showed that:

• neither the neither the Carroll SCFCarroll SCF (Counterexample with 7 voters and 8 candidates)(Counterexample with 7 voters and 8 candidates)

• nor the nor the Young SCFYoung SCF (Counterexample with 37 voters and 5 candidates)(Counterexample with 37 voters and 5 candidates)

is homogeneous... is homogeneous... BUTBUT they can be made homogeneous by: they can be made homogeneous by:

q

qVc,C,Vc,C,

)Score(lim )(*Score

q

J. Rothe, H. Spakowski & J. Vogel (TOCS, 2002):J. Rothe, H. Spakowski & J. Vogel (TOCS, 2002):

In the homogeneous case, Carroll*WinnerCarroll*Winner and Carroll*RankingCarroll*Ranking are efficiently solvable by a linear program.


Edsger Dijkstra

OutlineOutline




Voting game: G = (w1, …, wn; q). Our notation:• N = {1, …, n} : set of players• w1, …, wn : weights of players• q : quota value.

Power Indices – Power Indices – Banzhaf [1965] and Shapley-Shubik [1954]Banzhaf [1965] and Shapley-Shubik [1954]

Banzhaf*(G,i) = how many of the 2n-1 subsets of N – {i} have total weight < q but ≥ q-wi?

Banzhaf(G,i) = Banzhaf*(G,i)/2n-1

(Probability that a randomly chosen coalition of players in N – {i} is not successful but player i will put them over the top.)

SS*(G,i) = in how many of the n! permutations of N is i pivotal, i.e., the players before it sum to less than q but player i puts them over the top.

SS(G,i) = SS*(G,i)/n!

3 3 4 q = 6

#P (Counting NP):f #P if there is a nondeterministic

polynomial-time Turing machine M such that

#P: standard “counting” version of NP.

Complexity Classes: Complexity Classes: PP [Simon/Gill, 1970s] and #P [Valiant, 1979]PP [Simon/Gill, 1970s] and #P [Valiant, 1979]

PP (Probabilistic Polynomial Time):L PP if there is a probabilistic

polynomial-time Turing machine that has acceptance probability greater than 50% precisely on the strings in L.

(Or… “on most paths.”)

A A A

x

M f(x) = 3

x L

A R AR R A

x L

(xΣ*)[ f(x) = number of accepting paths of M on input x].

Known Results about PP:• NP PP• PH PPP [Toda, 1991]• PPP = P#P [BBS, 1986]• PNP PP [BHW, 1991]• PPP = PP [FR, 1996]

Known Results on PP:• NP PP• PH PPP [Toda, 1991]• PPP = P#P [BBS, 1986]• PNP PP [BHW, 1991]• PPP PP [BRS, 1992]

||||

||||

““Hardest” Problems for Classes: CompletenessHardest” Problems for Classes: Completeness

#P-completeness:#P-complete?

Multiple notions!

PP-completeness:

A B

f

f

Complete, yes. But how

complete?

•Polynomial-time Many-One Reducibility: A B (fFP)(xΣ*)[xA f(x)B].•B is B is hard for class hard for class CC if if (ACC)[A B].•B is C-C-complete if B is in C C and is hard for and is hard for C C ..

PPmm

PPmm

““Hardest” Problems for Function Classes: Hardest” Problems for Function Classes: CompletenessCompleteness

φ(x)

Definition:1. [Krentel, 1988] A function f:Σ*N metric

reduces to a function g:Σ*N if there exist two FP functions, φ and ψ, such that

(xΣ*)[ f(x) = ψ( x, g( φ(x) ) ) ].

2. [Zankó, 1991] A function f:Σ*N many-one reduces to a function g:Σ*N if there exist two FP functions, φ and ψ, such that

(xΣ*)[ f(x) = ψ( g( φ(x) ) ) ].

3. [Simon, 1975] A function f:Σ*N parsimoniously reduces to a function g:Σ*N if there exists an FP function φ such that (xΣ*)[ f(x) = g(φ(x)) ].

g ψ f(x)

x

φ(x) g ψ f(x)

φ(x) g f(x)

• Examples:• #SAT is #P-parsimonious-complete [L. Valiant, 1979].• SS* is #P-metric-complete [X. Deng & C.Papadimitriou,

1994].

#P-metric-complete

#P-many-one-complete

#P-parsimonious-complete

““Hardest” Problems for Function Classes: Hardest” Problems for Function Classes: CompletenessCompleteness

• Reductions for function classes• parsimonious• many-one• metric.

• Each defines a completeness notion: f is #P-foo-complete if• f #P, and• each #P function foo-reduces to f.

Dunno ’bout you, but I’m completely

lost!

Results for Computing Power IndicesResults for Computing Power Indices

X. Deng & C. Papadimitriou (1994):X. Deng & C. Papadimitriou (1994):SS* is #P-metric-complete.

Prasad & Kelly (1990)+Hunt, Marathe, Radhakrishnan & Stearns (1998):Prasad & Kelly (1990)+Hunt, Marathe, Radhakrishnan & Stearns (1998):

Banzhaf* is #P-parsimonious-complete.

P. Faliszewski & L. Hemaspaandra (2008):P. Faliszewski & L. Hemaspaandra (2008):• SS* is #P-many-one-complete.• SS* is not #P-parsimonious-

complete.

No, We No, We Can‘t!Can‘t!

NoNo, , We We Can‘t!Can‘t!

Question: Question:


(Can we improve this to #P-many-one-(Can we improve this to #P-many-one-completeness?) completeness?)

Aha… that complete is SS* for #P!#P-metric-complete

#P-many-one-complete

#P-parsimonious-completeSS

*

Question: Question:


(Can we improve this to #P-parsimonious-(Can we improve this to #P-parsimonious-completeness?) completeness?)

P. Faliszewski & L. Hemaspaandra (2008):P. Faliszewski & L. Hemaspaandra (2008):• SS* is #P-many-one-complete.

Power-Index Comparison is PP-CompletePower-Index Comparison is PP-Complete

20 papers

20 papers

50 papers $2M $5M $2M

4 papers$10M

Harvard University Money University

Where will I have more

(local) power?

Aha! Clearly, I will have more (local) power at

Money University! But how else can I justify this choice?Recall: Voting game G = (w1, …, wn; q).

Power-Index Comparison is PP-CompletePower-Index Comparison is PP-Complete

PowerCompare-PIPowerCompare-PI(where (where PI PI is either is either Banzhaf* or SS*))

Instance: Two voting games, G = (w1, …, wn; q) and G’ = (w’1, …, w’n; q’), and an integer i, 1 ≤ i ≤ n.Question: Is it true that PI( G, i ) > PI( G’, i )?

P. Faliszewski & L. Hemaspaandra (2008):P. Faliszewski & L. Hemaspaandra (2008):• PowerCompare-PowerCompare-BanzhafBanzhaf** is PP-complete.• PowerCompare-PowerCompare-SSSS** is PP-complete.

Proof IdeaProof Idea• PowerCompare-PowerCompare-BanzhafBanzhaf** is PP-complete: follows from

• Prasad & Kelly‘s result thatPrasad & Kelly‘s result that Banzhaf* is #P-parsimonious-complete and

• the fact that if f is any #P-parsimonious-complete function then the set Compare-fCompare-f = { (x,y) | x,yΣ* and f(x) > f(y)} is PP-complete.

• PowerCompare-PowerCompare-SSSS** is PP-complete: needs different arguments, since SS* is not #P-parsimonious-complete.

Further Results on Weighted Voting Further Results on Weighted Voting GamesGames

E. Elkind, L. Goldberg, P. Goldberg & M. Wooldridge (2007):E. Elkind, L. Goldberg, P. Goldberg & M. Wooldridge (2007):• Studied the complexity of other aspects of weighted voting

games:• TheThe core core• The least coreleast core• The nucleolusnucleolus

Notions that help finding coalitions that are stableand have fair imputations

• Provided:• Polynomial-time algorithms• NP-hardness results• Pseudopolynomial-time algorithms• Approximation algorithms If we are done with

weighted voting now, we could start

cutting a cake!


Edsger Dijkstra

OutlineOutline









a






• An allocation is envy-freeenvy-free if every agent is at least as happy with its share as with any of the other agents‘ shares.

Formally:

• An allocation is Pareto optimalPareto optimal if it is not Pareto-dominated by any other allocation. That is, for no allocation does it hold that

)()(, aPbPAba a

QPAbQPAa ba Q

P

P

a

Y. Chevaleyre, U. Endriss, S. Estivie & N. Maudet (2004) andY. Chevaleyre, U. Endriss, S. Estivie & N. Maudet (2004) and

P. Dunne, M. Wooldridge & M Laurence (2005):P. Dunne, M. Wooldridge & M Laurence (2005):

Welfare Opimization Welfare Opimization andand Welfare Improvement Welfare Improvement are NP-are NP-complete.complete.

Some Complexity Results inSome Complexity Results inMultiagent Resource AllocationMultiagent Resource Allocation

Definition: Definition: Let be a given resource allocation setting, and let be a given allocation. The utilitarian social welfare of is defined as the sum of individual utilities:

),,( URA

Aa

a PuPusw )()(

Welfare OpimizationWelfare OpimizationInstance: Resource allocation setting and a rational .Question: Does there exist an allocation such that ?

),,( URAP

k

kPusw )(

Welfare ImprovementWelfare ImprovementInstance: Resource allocation setting and an allocation .Question: Does there exist another allocation such that ?

Q)()( PuswQusw

),,( URA P

PP

Y. Chevaleyre, U. Endriss, S. Estivie & N. Maudet (2004) andY. Chevaleyre, U. Endriss, S. Estivie & N. Maudet (2004) and

P. Dunne, M. Wooldridge & M Laurence (2005):P. Dunne, M. Wooldridge & M Laurence (2005):

Pareto Optimality Pareto Optimality is coNP-complete.is coNP-complete.

Some Complexity Results inSome Complexity Results inMultiagent Resource AllocationMultiagent Resource Allocation

Envy-FreenessEnvy-FreenessInstance: Resource allocation setting .Question: Does there exist an envy-free allocation?

),,( URA

Pareto OptimalityPareto OptimalityInstance: Resource allocation setting and an allocation .Question: Is Pareto optimal?

),,( URA PP

S. Bouveret & J. Lang (2005):S. Bouveret & J. Lang (2005):• Envy-Freeness Envy-Freeness is NP-complete.is NP-complete.• For problems that combine For problems that combine Pareto Optimality Pareto Optimality andand

Envy-FreenessEnvy-Freeness

they prove complexity results ranging from NP-they prove complexity results ranging from NP-completeness up to completeness for the second level completeness up to completeness for the second level of the PH.of the PH.

A Conjecture A Conjecture from the MARA Survey by Chevaleyre et al.from the MARA Survey by Chevaleyre et al.

Exact Welfare OptimizationExact Welfare OptimizationInstance: Resource allocation setting and a rational .Question: Is the maximum utilitarian social welfare exactly equal

to , i.e., is it true that

?

),,( URA k

kPPusw }:)({ allocation an is maxk

Chevaleyre, Dunne, Endriss, Lang, Lemaitre, Maudet,Chevaleyre, Dunne, Endriss, Lang, Lemaitre, Maudet,

Padget, Phelps, Rodriguez-Aguilar & Sousa (2005):Padget, Phelps, Rodriguez-Aguilar & Sousa (2005):

ConjectureConjecture: : Exact Welfare OptimizationExact Welfare Optimization is DP-complete. is DP-complete.

Examples of DP-complete problems from graph theory:• Exact-4-ColorExact-4-Color: Given a graph, is its chromatic number exactly 4?Rothe (2003):Rothe (2003): Exact-4-Color Exact-4-Color is DP-complete.is DP-complete.

Exact-4-ColorExact-4-ColorExact-4-ColorExact-4-Color




?

),,( URA k





Examples of DP-complete problems from graph theory:• Exact-4-ColorExact-4-Color: Given a graph, is its chromatic number exactly 4?Rothe (2003):Rothe (2003): Exact-4-Color Exact-4-Color is DP-complete.is DP-complete.• Min-3-UncolorMin-3-Uncolor: Given a graph, decideif it is not 3-colorable but removing evenjust one vertex makes it 3-colorable?Cai & Meyer (1987): Cai & Meyer (1987): Min-3-Uncolor Min-3-Uncolor is DP-complete.is DP-complete.

Min-3-UncolorMin-3-UncolorMin-3-UncolorMin-3-Uncolor




?

),,( URA k





Examples of DP-complete problems from graph theory:• Exact-4-ColorExact-4-Color: Given a graph, is its chromatic number exactly 4?Rothe (2003):Rothe (2003): Exact-4-Color Exact-4-Color is DP-complete.is DP-complete.• Min-3-UncolorMin-3-Uncolor: Given a graph, decideif it is not 3-colorable but removing evenjust one vertex makes it 3-colorable?Cai & Meyer (1987): Cai & Meyer (1987): Min-3-Uncolor Min-3-Uncolor is DP-complete.is DP-complete.

Min-3-UncolorMin-3-Uncolor

The Boolean Hierarchy over NPThe Boolean Hierarchy over NP

Defining the Boolean Hierarchy over NP:Defining the Boolean Hierarchy over NP:

• Level 0: Level 0: PP (deterministic polynomial time)(deterministic polynomial time)• Level 1 has two classes:Level 1 has two classes:

• NPNP (nondeterministic polynomial time)(nondeterministic polynomial time)• coNP coNP (the class of complements of problems in NP)(the class of complements of problems in NP)

• Level 2 has two classes:Level 2 has two classes:• DP DP == { A-B : : A, BA, B NP } } (Difference-NP)(Difference-NP)• coDP coDP (the class of complements of problems in DP)(the class of complements of problems in DP)

• Level k has two classes:Level k has two classes:• BH(k) BH(k) == { L : L is the nested difference ofthe nested difference of kk NP NP setssets } } • coBH(k)coBH(k)

• BH BH is the union of all these levels.is the union of all these levels.

The levels of the BH capture the idea of The levels of the BH capture the idea of „„hardware over NP.“hardware over NP.“

Hey! That reminds me of the polynomial

hierarchy!

But what is DP?Desparate Prayers?

It seems to me that DP is full of

Difficult Problems.

The Boolean Hierarchy over NPThe Boolean Hierarchy over NP

Summary: A Landscape of Complexity ClassesSummary: A Landscape of Complexity Classes

P

NP coNP

DP coDP

BH

PNP||||

PH

PNP

NPNP coNPNP

P

NP coNP

DP coDP

BH

PPP=P#P

PP=PPP||||

PSPACESo what do you do

in complexity theory? I can’t

remember… It was so complicated…

•Studying hierarchies:Studying hierarchies:•Boolean HierarchyBoolean Hierarchy•Polynomial HierarchyPolynomial Hierarchy

•Proving problemsProving problems complete forcomplete for complexity classescomplexity classes

•Probabilistic andProbabilistic and counting classescounting classes

coNP• Pareto Pareto

OptimizatioOptimizationn

DP

• Min-3-UncolorMin-3-Uncolor• Exact-4-ColorExact-4-Color• Exact Welfare Exact Welfare

Optimization ?Optimization ?????

PNP||||

• Carroll Carroll WinnerWinner

• Young Young WinnerWinner

• Kemeny Kemeny WinnerWinner

PP=PPP||||

• PowerCompaPowerCompare-re-

• Banzhaf*Banzhaf*• SS*SS*

NPNP

• Pareto OptimalPareto Optimal Envy-FreenessEnvy-Freeness

NP

• Carroll ScoreCarroll Score• Manipulation for Manipulation for

some voting some voting systemssystems

• Envy-FreenessEnvy-Freeness• Welfare Welfare

OptimizationOptimization• Welfare Welfare

ImprovementImprovement

Any Literature Recommendations?Any Literature Recommendations?What if I can read only German?What if I can read only German?

... and a Call for Papers... and a Call for Papers„„Logic and Complexity within Computational Social Choice“Logic and Complexity within Computational Social Choice“

To appear as a special issue of To appear as a special issue of Mathematical Logic QuarterlyMathematical Logic QuarterlyEdited by Paul Goldberg and Jörg RotheEdited by Paul Goldberg and Jörg Rothe

Deadline: September 15, 2008Deadline: September 15, 2008

Thank you!Thank you!

I hope they won’t ask any questions!

computational complexity for social choice theorists jörg rothe comsoc 2008, liverpool, uk

Documents

complexity of problems

intractable problems

hardest problems

computational complexity

class c

funny names

c complete

pmpmpmpm pmpmpmpm slide