Fixed-size Minimax for Committee Elections:

Approximation and Local Search Heuristics

Rob LeGrandWashington University in St. Louis

legrand@cse.wustl.edu

Evangelos MarkakisUniversity of Toronto

vangelis@cs.toronto.edu

Aranyak MehtaIBM Almaden Research Center

mehtaa@us.ibm.com

COMSOC ’066 December 2006

2

Electing a committee from approval ballots

11110 00011

00111

0000110111

01111

•What’s the best committee of size k = 2?

approves ofcandidates

4 and 5n = 5 candidates

m = 6 ballots

3

Sum of Hamming distances

11110 00011

00111

0000110111

01111 110004 5

2 4

4 3 sum = 22

k = 2 winners

4

Fixed-size minisum

11110 00011

00111

0000110111

01111 00011

•Minisum elects winner set with smallest sumscore•Easy to compute (pick candidates with most approvals)

2 1

4 0

2 1 sum = 10

k = 2 winners

5

Maximum Hamming distance

11110 00011

00111

0000110111

01111 000112 1

4 0

2 1 sum = 10max = 4

k = 2 winners

6

Fixed-size minimax

•Minimax elects winner set with smallest maxscore•Harder to compute?

11110 00011

00111

0000110111

01111 001102 1

2 2

2 3 sum = 12max = 3

k = 2 winners

[Brams, Kilgour & Sanver, ’04]

7

Complexity

Endogenous minimax

= EM = BSM(0, n)

Bounded-size minimax

= BSM(k1, k2)

Fixed-size minimax

= FSM(k) = BSM(k, k)

NP-hard

[Frances & Litman, ’97]

NP-hard

(generalization of EM)

?

8

Complexity

Endogenous minimax

= EM = BSM(0, n)

Bounded-size minimax

= BSM(k1, k2)

Fixed-size minimax

= FSM(k) = BSM(k, k)

NP-hard

[Frances & Litman, ’97]

NP-hard

(generalization of EM)

NP-hard

(this paper)

9

Approximability

Endogenous minimax

= EM = BSM(0, n)

Bounded-size minimax

= BSM(k1, k2)

Fixed-size minimax

= FSM(k) = BSM(k, k)

has a PTAS*

[Li, Ma & Wang, ’99]

no known PTAS;no known constant-

factor approx.

no known PTAS;no known constant-

factor approx.

* Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε

10

Approximability

Endogenous minimax

= EM = BSM(0, n)

Bounded-size minimax

= BSM(k1, k2)

Fixed-size minimax

= FSM(k) = BSM(k, k)

has a PTAS*

[Li, Ma & Wang, ’99]

no known PTAS;

has a 3-approx.

(this paper)

no known PTAS;

has a 3-approx.

(this paper)

* Polynomial-Time Approximation Scheme: algorithm with approx. ratio 1 + ε that runs in time polynomial in the input and exponential in 1/ε

11

Approximating FSM

00111

00001

10111

01111

00011

11110

00111

k = 2 winners

choosea ballot

arbitrarily

12

Approximating FSM

00111

00001

10111

01111

00011

11110

0010100111coerce to

size k

k = 2 winners

choosea ballot

arbitrarily

outcome =k-completed ballot

13

Approximation ratio ≤ 3

00111

00001

10111

01111

00011

11110

00110

2

2

1

3

2

2

≤ OPT

optimalFSM set

OPT = optimal maxscore

14

Approximation ratio ≤ 3

00111

00001

10111

01111

00011

11110

00110 00111

2

2

1

3

2

2

1

≤ OPT ≤ OPT

optimalFSM set

chosenballot

OPT = optimal maxscore

15

Approximation ratio ≤ 3

00111

00001

10111

01111

00011

11110

00110 00111 00011

2

2

1

3

2

2

1 1

≤ OPT ≤ OPT ≤ OPT

≤ 3·OPT

optimalFSM set

chosenballot

k-completedballot

OPT = optimal maxscore (by triangle inequality)

16

Better in practice?

• So far, we can guarantee a winner set no more than 3 times as bad as the optimal.– Nice in theory . . .

• How can we do better in practice?– Try local search

17

Local search approach for FSM

1. Start with some c {0,1}n of weight k

010014

18

Local search approach for FSM

1. Start with some c {0,1}n of weight k

2. In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result

01001

11000 10001

01100

01010 00011

001014

44

4

5

4

4

19

Local search approach for FSM

1. Start with some c {0,1}n of weight k

2. In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result

010104

20

Local search approach for FSM

1. Start with some c {0,1}n of weight k

2. In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result

010104

21

Local search approach for FSM

1. Start with some c {0,1}n of weight k

2. In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result

3. Repeat step 2 until maxscore(c) is unchanged n times

4. Take c as the solution

01010

11000 10010

01100

01001 00011

001104

44

4

5

3

4

22

Local search approach for FSM

1. Start with some c {0,1}n of weight k

2. In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result

3. Repeat step 2 until maxscore(c) is unchanged n times

4. Take c as the solution

001103

23

Specific FSM heuristics

• Two parameters:– where to start vector c:

1. a fixed-size-minisum solution

2. a k-completion of a ballot (3-approx.)

3. a random set of k candidates

4. a k-completion of a ballot with highest maxscore– radius of neighborhood r: 1 and 2

24

Heuristic evaluation

• Real-world ballots from GTS 2003 council election• Found exact minimax solution• Ran each heuristic 5000 times• Compared exact minimax solution with heuristics to find

realized approximation ratios– example: 15/14 = 1.0714

• maxscore of solution found = 15• maxscore of exact solution = 14

• We also performed experiments using ballots generated according to random distributions (see paper)

25

Average approx. ratios found

radius = 1 radius = 2fixed-size minimax

1.0012 1.0000

3-approx. 1.0017 1.0000

random set

1.0057 1.0000

highest-maxscore

1.0059 1.0000

performance on GTS ’03 election data

n = 24 candidates, k = 12 winners, m = 161 ballots

26

Largest approx. ratios found

radius = 1 radius = 2fixed-size minimax

1.0714 1.0000

3-approx. 1.0714 1.0000

random set

1.0714 1.0000

highest-maxscore

1.0714 1.0000

performance on GTS ’03 election data

n = 24 candidates, k = 12 winners, m = 161 ballots

27

Conclusions from all experiments

• All heuristics perform near-optimally– highest ratio found: 1.2– highest average ratio < 1.04

• When radius is larger, performance improves and running time increases

• The fixed-size-minisum starting point performs best overall (with our 3-approx. a close second)

28

Manipulating FSM

00110 00011

00111

0000110111

01111 00011

•Voters are sincere

•Another optimal solution: 00101

2 1

2 0

2 1

max = 2

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Manipulating FSM

11110 00011

00111

0000110111

01111 00110

•A voter manipulates and realizes ideal outcome

2 1

2 2

2 3

00110

0

max = 3

30

Nonmanipulable “FSM”?

Electing a set found using our 3-approximation for FSM gives a nonmanipulable procedure:

• For the voters whose ballots are not chosen, voting insincerely cannot affect the outcome

• For the voter whose ballot is chosen, the outcome will be one of the sets of size k closest to the voter’s wishes

31

Conclusions

• BSM and FSM are NP-hard• Both can be approximated with ratio 3• Polynomial-time local search heuristics perform well

in practice– some retain ratio-3 guarantee

• Exact FSM can be manipulated• Our 3-approximation for FSM is nonmanipulable

32

Future work

• Investigate weighted version of minimax [Brams, Kilgour & Sanver, ’06]

• What is the best approximation ratio for FSM achievable in polynomial time? (Is there a PTAS?)

• What is the nonmanipulable FSM approximation algorithm with the best ratio?

Thanks!