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On the Fixed-Parameter Tractability of Composition-Consistent - - PowerPoint PPT Presentation

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On the Fixed-Parameter Tractability of Composition-Consistent Tournament Solutions Hans Georg Seedig COMSOC 2010 August 14, 2010 Joint work with Felix Brandt and Markus Brill Overview Tournaments Components and Decomposition

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SLIDE 1

On the Fixed-Parameter Tractability

  • f Composition-Consistent

Tournament Solutions

Hans Georg Seedig

COMSOC 2010 August 14, 2010 Joint work with Felix Brandt and Markus Brill

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SLIDE 2

Overview

  • Tournaments
  • Components and Decomposition
  • Tournament Solutions
  • Composition Consistency
  • Parametrized Complexity
  • Fixed-parameter tractability
  • Algorithm
  • Experiments

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SLIDE 3

Tournaments

  • T=(A, ≻) is a tournament
  • A is a finite set of candidates or alternatives
  • ≻ is an asymmetric and complete binary

relation on the alternatives

  • a ≻ b means ‘a dominates b’ or ‘a is

preferred over b’

  • pairwise majority outcome of an election
  • ≻ may be cyclic
  • Corresponds to complete oriented

graph

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SLIDE 4
  • Alternatives in a tournament form a component if they

bear the same relationship to all outside alternatives

Components in Tournaments

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SLIDE 5
  • Alternatives in a tournament form a component if they

bear the same relationship to all outside alternatives

Components in Tournaments

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SLIDE 6
  • Alternatives in a tournament form a component if they

bear the same relationship to all outside alternatives

Components in Tournaments

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SLIDE 7

Decompositions

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  • A graph can be decomposed into components
  • A decomposition of T=(A, ≻) is a set of pairwise disjoint

components {B₁,B₂,...,Bk} such that ∪Bi = A

  • The summary of T w.r.t this decomposition is the

tournament on the components Ť, induced by T.

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SLIDE 8

Decomposition Tree

  • There is a unique minimal decomposition
  • A component may be decomposable again
  • Represent this recursive decompositions as a

decomposition tree

  • The decomposition degree δ is the maximum degree in

the decomposition tree

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SLIDE 9

c

Example: Decomposition Tree

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a b g f e d

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SLIDE 10

f

Example: Decomposition Tree

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b c d e g a

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SLIDE 11

f

Example: Decomposition Tree

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b c d e g a

a,b,c,d,e,f,g

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SLIDE 12

f

Example: Decomposition Tree

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b c d e g a

a,b,c,d,e,f,g f,c a,d,e,g b

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SLIDE 13

f

Example: Decomposition Tree

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b c d e g a

a,b,c,d,e,f,g f,c a,d,e,g b d,e

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SLIDE 14

f

Example: Decomposition Tree

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b c d e g a

a,b,c,d,e,f,g f,c a,d,e,g b a g d,e

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SLIDE 15

f

Example: Decomposition Tree

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b c d e g a

a,b,c,d,e,f,g f,c a,d,e,g b f c a g d,e d e

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SLIDE 16

f

Example: Decomposition Tree

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b c d e g a

a,b,c,d,e,f,g f,c a,d,e,g b f c a g d,e d e

  • Decomposition degree δ is
  • max. degree in decomp. tree

δ=3

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SLIDE 17

Tournament Solutions

  • Given a tournament, what is the set of winners?
  • Intuitively easy if one alternative c dominates all others
  • c is a Condorcet winner
  • does not exist in most tournaments
  • A tournament solution S returns a non-empty subset of

A, i.e., S(T)⊆A

  • Many solution concepts have been proposed in the past
  • Axiomatic approach: Do they have desirable properties?

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SLIDE 18

Zoo of Tournament Solutions

  • Copeland set
  • Slater set
  • Banks set
  • Uncovered Set
  • Minimal Covering Set (MC)
  • Bipartisan Set (BP)
  • TEQ

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  • Many tournament solutions are computationally hard
  • Slater, Banks and TEQ are NP-hard. MC and BP are in P but existing

algorithms rely on linear programming and are thus rather inefficient.

  • All of these except Copeland and Slater satisfy composition-consistency.
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Composition-Consistency

  • A tournament solution S is composition-consistent if it

chooses the ‘best’ alternatives from the ‘best’ components.

  • Formally: S is composition-consistent if for all T, Ť summary
  • f T w.r.t. some decomposition {B1,...,Bk}

S(T)=∪i∈S(Ť)S(T|Bi)

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SLIDE 20

Fixed-Parameter Tractability

  • Use parametrized complexity to analyze whether the

hardness of a problem depends on the size of a certain parameter

  • Consider a problem with parameter k fixed-parameter

tractable (FPT) if there is an algorithm that solves it in time f(k)⋅poly(InputLength) where f is independent of the input length

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SLIDE 21

Algorithm

  • 1. Compute the decomposition tree
  • 2. Recursively compute tournament

solution on components

  • Decomposition tree computable in linear time!
  • Follows from results by McConnell and de Montgolfier (2005);

Capelle et al. (2002) on modular decomposition of directed graphs

  • Number of tournaments to solve is bounded by |A|-1
  • Size of the largest tournament to solve equals the

decomposition degree

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Algorithm

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SLIDE 22

Main Result

Given composition-consistent tournament solution S where computing S(T) with |T|≤k takes time ≤f(k) Then, S(T) can be computed in O(|T|2)+f(δ(T))⋅(|T|-1) Corollary Computing S(T) is fixed-parameter tractable w.r.t. δ(T).

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compute decomposition tree worst-case time for solving a tournament

  • max. no.
  • f tournaments
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SLIDE 23

Experiments

  • Generate majority tournaments according to voting

models

  • Noise model:

Voters give “correct” ranking of each pair of alternatives with probability p > ½

  • Spatial model: Alternatives and voters are located in [0,1]d.

Preferences according to Euclidian distances between voters and alternatives.

  • a ≻ b iff a majority prefers a to b

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SLIDE 24

Noise model with p=0.55

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 normalized decomposition degree number of voters 10 candidates 50 candidates 100 candidates 150 candidates 200 candidates

δ/|A|

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SLIDE 25

Spatial model with d=2

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 normalized decomposition degree number of voters 10 candidates 50 candidates 100 candidates 150 candidates 200 candidates

corresponds to a speed-up of ~2.5⋅1020 if S(T) takes time 2|T| δ/|A|

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SLIDE 26

Spatial model with d=20

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 normalized decomposition degree number of voters 10 candidates 50 candidates 100 candidates 150 candidates 200 candidates

δ/|A|

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SLIDE 27

Conclusion

  • Exploiting composition-consistency can lead to dramatical

speed ups in algorithms for tournament solutions

  • All tournament solutions satisfying composition-

consistency are fixed-parameter tractable w.r.t. the decomposition degree

  • δ=O(logk |A|) for some k allows polynomial-time

algorithms for tournament solutions that in general only admit algorithms of time O(2n)

  • Future work
  • Measure positive effect by actual computation of composition-

consistent tournament solutions.

  • Use parallelization and lookup tables.

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