Consider a fixed voting rule R. In the Possible President problem, we are given an election where the candidates are partitioned into parties, and the problem is to determine if, given a party P, it is possible for every party to nominate a candidate such that the nominee from P is a winner of the election that is obtained by restricting the votes to the nominated candidates. In the Necessary President problem, we would like to find a nominee who wins no matter who else is nominated. In this talk, we explore the complexity of these problems, which can be thought of as the two natural extremes of the party nomination problem, with an emphasis on a parameterized perspective and algorithms on structured profiles.

1. Neeldhara Misra, IIT Gandhinagar
Dominik Peters, Carnegie-Mellon University
2019 Conference on
Algorithmic Decision Theory
Parameterized Complexity of Party Nominations

2. AB
C P
Q
R
X
Y
Z
S
T
Candidates

3. AB
C P
Q
R
X
Y
Z
S
T
B C A S P Q YZ TRX
Candidates
Votes

4. AB
C P
Q
R
X
Y
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates

5. AB
C P
Q
R
X
Y
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates

6. AB
C P
Q
R
X
Y
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates
The candidates are partitioned into “parties”.

7. AB
C P
Q
R
X
Y
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates
Every party nominates a candidate.

8. AB
C P
R
X
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates
Every party nominates a candidate.
Q
Y

9. AB
C P
R
X
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates
Every party nominates a candidate.
Q
Y
The votes are “projected” on the nominees.

10. AB
C P
R
X
Z
S
T
Votes
B C A S P Q YZ TRX
Candidates
Every party nominates a candidate.
Q
Y
The winner is declared based on the plurality voting rule.

11. Assuming complete knowledge about the votes,
how do parties select their nominees?
The Problem

12. An Example
BA X
B AX
B AX
6
4
1
A BCandidates
Votes
X

13. An Example
BA X
B AX
B AX
6
4
1
B XCandidates
Votes
A

14. An Example
BA X
B AX
B AX
6
4
1
XCandidates
Votes
A B

15. Assuming complete knowledge about the votes,
how do parties select their nominees?
The Problem

16. Assuming complete knowledge about the votes,
how do parties select their nominees?
The Problem
What do the parties know about
other nominees?

17. Plurality

18. If we know who the other parties are nominating,
it is easy to “evaluate” a candidate in our party.
Plurality

19. If we know who the other parties are nominating,
it is easy to “evaluate” a candidate in our party.
Plurality

20. If we know who the other parties are nominating,
it is easy to “evaluate” a candidate in our party.
Plurality

21. Plurality
If we know who the other parties are nominating,
it is easy to “evaluate” a candidate in our party.

22. A more natural scenario
We have no idea who the other nominees are.

23. A more natural scenario
We have no idea who the other nominees are.
The Optimist’s Question
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?

24. A more natural scenario
We have no idea who the other nominees are.
The Optimist’s Question
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?
The Pessimist’s Question
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?

25. A more natural scenario
We have no idea who the other nominees are.
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?
The Pessimist’s Question
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?
Necessary President

26. A more natural scenario
We have no idea who the other nominees are.
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?
Necessary President
Possible President

27. Known Results
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?
Necessary President
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

28. Known Results
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?
Necessary President
co-NP complete even when the size of the largest party is two.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

29. Known Results
Do we have a superstar candidate who ensures a party win,
irrespective of who is nominated from the other parties?
Necessary President
co-NP complete even when the size of the largest party is two.
polynomial-time when the profiles are single-peaked.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

30. Known Results
Possible President
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

31. Known Results
Possible President
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?
NP-complete even when the size of the largest party is two.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

32. Known Results
Possible President
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?
NP-complete even when the size of the largest party is two.
NP-complete also when the profiles are 1D-Euclidean.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

33. Known Results
Possible President
Do we have a promising candidate who makes the party win
in at least one of the many possible parallel universes?
NP-complete even when the size of the largest party is two.
NP-complete also when the profiles are 1D-Euclidean.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016
(a subclass of single-peaked & single-crossing profiles)

34. Our Contributions

35. Possible President
Our Contributions

36. Possible President
NP-complete also when the profiles are 1D-Euclidean.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016
Our Contributions
NP-complete even when the size of the largest party is two.

37. Possible President
NP-complete even when the size of the largest party is two,
Our Contributions
NP-complete also when the profiles are 1D-Euclidean.and the

38. Possible President
NP-complete even when the size of the largest party is two,
Our Contributions
NP-complete also when the profiles are 1D-Euclidean.
(A stronger hardness result.)
and the

39. Possible President
NP-complete even when the size of the largest party is two,
Our Contributions
NP-complete also when the profiles are 1D-Euclidean.
(A stronger hardness result.)
and the
XP and W[2]-hard parameterized by the number of parties.

40. Possible President
NP-complete even when the size of the largest party is two,
Our Contributions
NP-complete also when the profiles are 1D-Euclidean.
(A stronger hardness result.)
and the
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.

41. Possible President
NP-complete even when the size of the largest party is two,
Our Contributions
NP-complete also when the profiles are 1D-Euclidean.
(A stronger hardness result.)
and the
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
(Parameterized Results)

42. Necessary President
Our Contributions

43. Necessary President
Our Contributions
co-NP complete even when the size of the largest party is two.
polynomial-time when the profiles are single-peaked.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016

44. Necessary President
Our Contributions
co-NP complete even when the size of the largest party is two.
polynomial-time when the profiles are single-peaked.
Piotr Faliszewski, Laurent Gourvès, Jérôme Lang, Julien Lesca, Jèrôme Monnot.
How hard is it for a party to nominate an election winner? AAAI 2016
polynomial-time when the profiles are single-crossing.

45. Talk Outline
Preliminaries: parameterized algorithms, restricted domains
High-level methodology
W[2]-hardness parameterized by #parties
Open Problems
Introduction

46. Talk Outline
Preliminaries: parameterized algorithms, restricted domains
High-level methodology
W[2]-hardness parameterized by #parties
Open Problems
Introduction

47. The Parameterized Paradigm
Beyond Worst-Case
Classical complexity: measure the performance of an algorithm
as a function of the input size.

48. Parameterized complexity: acknowledge the presence of additional structure,
which manifests as a secondary measurement — a parameter.
The Parameterized Paradigm
Beyond Worst-Case

49. Parameterized complexity: acknowledge the presence of additional structure,
which manifests as a secondary measurement — a parameter.
🎯 Design algorithms that restrict the combinatorial explosion
to a function of the parameter.
The Parameterized Paradigm
Beyond Worst-Case

50. Parameterized complexity: acknowledge the presence of additional structure,
which manifests as a secondary measurement — a parameter.
🎯 Design algorithms that restrict the combinatorial explosion
to a function of the parameter.
The Parameterized Paradigm
Beyond Worst-Case
f(k)p(n)
Input size
Parameter
fixed-parameter tractability

51. ⚠ W-hardness: a framework for arguing the
likely non-existence of FPT algorithms
for parameterized problems
The Parameterized Paradigm
Beyond Worst-Case
f(k)p(n)
Input size
Parameter
fixed-parameter tractability

52. ⚠ W-hardness: a framework for arguing the
likely non-existence of FPT algorithms
for parameterized problems
The Parameterized Paradigm
Beyond Worst-Case
😰
Hard
problem
X
FPT-Reductions

53. ⚠ W-hardness: a framework for arguing the
likely non-existence of FPT algorithms
for parameterized problems
The Parameterized Paradigm
Beyond Worst-Case
😰
Hard
problem
X
Runs in FPT time ● Preserves the parameter ● Maintains equivalence
FPT-Reductions

54. Talk Outline
Preliminaries: parameterized algorithms, restricted domains
High-level methodology
W[2]-hardness parameterized by #parties
Open Problems
Introduction

55. Possible President
High Level Methodology
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the

56. Possible President
High Level Methodology
Reduction from “Linear” SAT aka LSAT
(a structured variation of SAT,
originally used in the context of geometric problems*)
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
* Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Matthew J. Katz, Joseph S. B. Mitchell,
Marina Simakov. Choice is Hard, ISAAC 2015

57. Possible President
High Level Methodology
Brute-force
(guess the nominee from each party)
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the

58. Possible President
High Level Methodology
FPT-reduction
(from a variant of Dominating Set,
also coming up in this talk)
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the

59. Possible President
High Level Methodology
Dynamic Programming
(updates along the 1D-Euclidean axis,
also appeals to “SP and SC aspects” of 1D-Euclidean profiles)
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the

60. Possible President
High Level Methodology
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
Necessary President
polynomial-time when the profiles are single-crossing.
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the

61. Possible President
High Level Methodology
XP and W[2]-hard parameterized by the number of parties.
FPT parameterized by number of parties on 1D-Euclidean profiles.
Necessary President
polynomial-time when the profiles are single-crossing.
Adversarial approach: guess a nominee + a rival candidate
(use a “block property” and reduce to a structured Hitting Set instance)
NP-complete even when the size of the largest party is two,
profiles are 1D-Euclidean.and the

62. Talk Outline
Preliminaries: parameterized algorithms, restricted domains
High-level methodology
W[2]-hardness parameterized by #parties
Open Problems
Introduction

63. Colourful Red-Blue Dominating Set
— hard parameterized by the “solution size”

64. Colourful Red-Blue Dominating Set
— hard parameterized by the “solution size”

65. Colourful Red-Blue Dominating Set
— hard parameterized by the “solution size”

66. Colourful Red-Blue Dominating Set
— hard parameterized by the “solution size”

67. Colourful Red-Blue Dominating Set
— hard parameterized by the “solution size”

68. Colourful Red-Blue Dominating Set
— hard parameterized by the “solution size”

69. W[2]-hardness of Possible President
(parameterized by #parties)

70. W[2]-hardness of Possible President
(parameterized by #parties)
Introduce a candidate for every red vertex;
and two special candidates p and q.

71. W[2]-hardness of Possible President
(parameterized by #parties)
Introduce a candidate for every red vertex;
and two special candidates p and q.
Parties. p,q are singletons.
The other parties correspond to color classes of the CRBDS instance.

72. Introduce a vote for every blue vertex with the ordering:
neighbours
non-neighbours
W[2]-hardness of Possible President
(parameterized by #parties)

73. W[2]-hardness of Possible President
(parameterized by #parties)

74. Also introduce n copies of two special votes:
W[2]-hardness of Possible President
(parameterized by #parties)

75. Also introduce n copies of two special votes:
W[2]-hardness of Possible President
(parameterized by #parties)

76. Also introduce n copies of two special votes:
W[2]-hardness of Possible President
(parameterized by #parties)
Ask if p is a possible president.

77. Also introduce n copies of two special votes:
W[2]-hardness of Possible President
(parameterized by #parties)
Ask if p is a possible president.
Answer: Yes if and only if the “other nominees”
correspond to a colourful red-blue dominating set.

78. Also introduce n copies of two special votes:
W[2]-hardness of Possible President
(parameterized by #parties)
Ask if p is a possible president.
To begin with, p and q tie at a score of n each.
p’s score is “locked in” at n.
Nominees from a dominating set
“block” q from acquiring any additional score.

79. Talk Outline
Preliminaries: parameterized algorithms, restricted domains
High-level methodology
W[2]-hardness parameterized by #parties
Open Problems
Introduction

80. Parameterized complexity when
parameterized by the number of voters?
Open Problems
🤔
Is Possible President parameterized by the number of parties FPT
on single-peaked or single-crossing domains?

81. Open Problems
🤔
Intermediate notions of incomplete information.
What if we have partial information about the other nominees,
served either in a stochastic fashion or
as a fixed fraction of the number of parties?

82. Thank You!