1. Fixed-size Minimax for Committee Elections: Approximation and Local Search Heuristics
COMSOC ’06
6 December 2006
Rob LeGrand
Washington University in St. Louis
Evangelos Markakis
University of Toronto
Aranyak Mehta
IBM Almaden Research Center

2. Electing a committee from approval ballots
approves of
candidates
4 and 5
11110
00011
n = 5 candidates
m = 6 ballots
01111
00111
10111
00001
What’s the best committee of size k = 2?

3. Sum of Hamming distances
11110
00011
k = 2 winners 2 4 4 5
01111
11000
00111 4 3
sum = 22
10111
00001

4. Fixed-size minisum Minisum elects winner set with smallest sumscore
11110
00011
k = 2 winners 4 2 1
01111
00011
00111 2 1
sum = 10
10111
00001
Minisum elects winner set with smallest sumscore
Easy to compute (pick candidates with most approvals)

5. Maximum Hamming distance
11110
00011
k = 2 winners 4 2 1
01111
00011
00111 2 1
sum = 10
max = 4
10111
00001

6. [Brams, Kilgour & Sanver, ’04]
Fixed-size minimax
[Brams, Kilgour & Sanver, ’04]
11110
00011
k = 2 winners 2 2 2 1
01111
00110
00111 2 3
sum = 12
max = 3
10111
00001
Minimax elects winner set with smallest maxscore
Harder to compute?

7. (generalization of EM)
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]
(generalization of EM) ?

8. (generalization of EM)
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]
(generalization of EM)
(this paper)

9. no known constant-factor approx.
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.
* 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 has a PTAS* has a 3-approx. no known PTAS;
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)
* 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 11110 k = 2 winners 00011 00111 00111 00001 choose
a ballot
arbitrarily
10111
01111

12. Approximating FSM 11110 k = 2 winners 00011 00111 coerce to size k
00101
00001
choose
a ballot
arbitrarily
10111
01111
outcome =
k-completed ballot

13. Approximation ratio ≤ 3 optimal FSM set 11110 2 00011 2 1 00111 00110
00001 2
10111 2
01111
≤ OPT
OPT = optimal maxscore

14. Approximation ratio ≤ 3 optimal FSM set chosen ballot 11110 2 00011 2
00111 1
00110
00111 3
00001 2
10111 2
01111
≤ OPT
≤ OPT
OPT = optimal maxscore

15. (by triangle inequality)
Approximation ratio ≤ 3
optimal
FSM set
chosen
ballot
k-completed
ballot
11110 2
00011 2 1
00111 1 1
00110
00111
00011 3
00001 2
10111 2
01111
≤ OPT
≤ OPT
≤ OPT
(by triangle inequality)
OPT = optimal maxscore
≤ 3·OPT

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
Start with some c  {0,1}n of weight k
01001 4

18. Local search approach for FSM
Start with some c  {0,1}n of weight k
In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result
11000
10001 5 4
01100
01001
00101 4 4 4
01010
00011 4 4

19. Local search approach for FSM
Start with some c  {0,1}n of weight k
In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result
01010 4

20. Local search approach for FSM
Start with some c  {0,1}n of weight k
In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result
01010 4

21. Local search approach for FSM
Start with some c  {0,1}n of weight k
In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result
Repeat step 2 until maxscore(c) is unchanged n times
Take c as the solution
11000
10010 5 4
01100
01010
00110 4 4 3
01001
00011 4 4

22. Local search approach for FSM
Start with some c  {0,1}n of weight k
In c, swap up to r 0-bits with 1-bits in such a way that minimizes the maxscore of the result
Repeat step 2 until maxscore(c) is unchanged n times
Take c as the solution
00110 3

23. Specific FSM heuristics
Two parameters:
where to start vector c:
a fixed-size-minisum solution
a k-completion of a ballot (3-approx.)
a random set of k candidates
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 =
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 = 2
fixed-size minimax
1.0012
1.0000
3-approx.
1.0017
random set
1.0057
highest-maxscore
1.0059
performance on GTS ’03 election data
n = 24 candidates, k = 12 winners, m = 161 ballots

26. Largest approx. ratios found
radius = 1
radius = 2
fixed-size minimax
1.0714
1.0000
3-approx.
random set
highest-maxscore
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 Voters are sincere Another optimal solution: 00101
00110
00011 2 2 1
01111
00011
00111 2 1
max = 2
10111
00001
Voters are sincere
Another optimal solution: 00101

29. Manipulating FSM A voter manipulates and realizes ideal outcome 00110
11110
00011 2 2 2 1
01111
00110
00111 2 3
max = 3
10111
00001
A voter manipulates and realizes ideal outcome

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. Both can be approximated with ratio 3
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!