TY - GEN
T1 - Gerrymandering on Graphs
T2 - 14th International Symposium on Algorithmic Game Theory, SAGT 2021
AU - Gupta, Sushmita
AU - Jain, Pallavi
AU - Panolan, Fahad
AU - Roy, Sanjukta
AU - Saurabh, Saket
N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - This paper studies gerrymandering on graphs from a computational viewpoint (introduced by Cohen-Zemach et al. [AAMAS 2018] and continued by Ito et al. [AAMAS 2019]). Our contributions are two-fold: conceptual and computational. We propose a generalization of the model studied by Ito et al., where the input consists of a graph on n vertices representing the set of voters, a set of m candidates C, a weight function wv: C→ Z+ for each voter v∈ V(G) representing the preference of the voter over the candidates, a distinguished candidate p∈ C, and a positive integer k. The objective is to decide if it is possible to partition the vertex set into k districts (i.e., pairwise disjoint connected sets) such that the candidate p wins more districts than any other candidate. There are several natural parameters associated with the problem: the number of districts (k), the number of voters (n), and the number of candidates (m). The problem is known to be NP-complete even if k= 2, m= 2, and G is either a complete bipartite graph (in fact K2,n, i.e., partitions of size 2 and n) or a complete graph. Moreover, recently we and Bentert et al. [WG 2021], independently, showed that the problem is NP-hard for paths. This means that the search for FPT algorithms needs to focus either on the parameter n, or subclasses of forest (as the problem is NP-complete on K2,n, a family of graphs that can be transformed into a forest by deleting one vertex). Circumventing these intractability results we successfully obtain the following algorithmic results. A 2 n(n+ m) O(1) time algorithm on general graphs.FPT algorithm with respect to k (an algorithm with running time 2 O(k)nO(1) ) on paths in both deterministic and randomized settings, even for arbitrary weight functions. Whether the problem is FPT parameterized by k on trees remains an interesting open problem. Our algorithmic results use sophisticated technical tools such as representative set family and Fast Fourier Transform based polynomial multiplication, and their (possibly first) application to problems arising in social choice theory and/or algorithmic game theory is likely of independent interest to the community.
AB - This paper studies gerrymandering on graphs from a computational viewpoint (introduced by Cohen-Zemach et al. [AAMAS 2018] and continued by Ito et al. [AAMAS 2019]). Our contributions are two-fold: conceptual and computational. We propose a generalization of the model studied by Ito et al., where the input consists of a graph on n vertices representing the set of voters, a set of m candidates C, a weight function wv: C→ Z+ for each voter v∈ V(G) representing the preference of the voter over the candidates, a distinguished candidate p∈ C, and a positive integer k. The objective is to decide if it is possible to partition the vertex set into k districts (i.e., pairwise disjoint connected sets) such that the candidate p wins more districts than any other candidate. There are several natural parameters associated with the problem: the number of districts (k), the number of voters (n), and the number of candidates (m). The problem is known to be NP-complete even if k= 2, m= 2, and G is either a complete bipartite graph (in fact K2,n, i.e., partitions of size 2 and n) or a complete graph. Moreover, recently we and Bentert et al. [WG 2021], independently, showed that the problem is NP-hard for paths. This means that the search for FPT algorithms needs to focus either on the parameter n, or subclasses of forest (as the problem is NP-complete on K2,n, a family of graphs that can be transformed into a forest by deleting one vertex). Circumventing these intractability results we successfully obtain the following algorithmic results. A 2 n(n+ m) O(1) time algorithm on general graphs.FPT algorithm with respect to k (an algorithm with running time 2 O(k)nO(1) ) on paths in both deterministic and randomized settings, even for arbitrary weight functions. Whether the problem is FPT parameterized by k on trees remains an interesting open problem. Our algorithmic results use sophisticated technical tools such as representative set family and Fast Fourier Transform based polynomial multiplication, and their (possibly first) application to problems arising in social choice theory and/or algorithmic game theory is likely of independent interest to the community.
KW - Gerrymandering
KW - Parameterized complexity
KW - Representative set
UR - http://www.scopus.com/inward/record.url?scp=85115853013&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-85947-3_10
DO - 10.1007/978-3-030-85947-3_10
M3 - Conference contribution
AN - SCOPUS:85115853013
SN - 9783030859466
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 140
EP - 155
BT - Algorithmic Game Theory - 14th International Symposium, SAGT 2021, Proceedings
A2 - Caragiannis, Ioannis
A2 - Hansen, Kristoffer Arnsfelt
PB - Springer Science and Business Media Deutschland GmbH
Y2 - 21 September 2021 through 24 September 2021
ER -