## Abstract

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 w_{v}: 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 K_{2}_{,}_{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 K_{2}_{,}_{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}^{)}n^{O}^{(}^{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.

Original language | English |
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Title of host publication | Algorithmic Game Theory - 14th International Symposium, SAGT 2021, Proceedings |

Editors | Ioannis Caragiannis, Kristoffer Arnsfelt Hansen |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 140-155 |

Number of pages | 16 |

ISBN (Print) | 9783030859466 |

DOIs | |

State | Published - 1 Jan 2021 |

Externally published | Yes |

Event | 14th International Symposium on Algorithmic Game Theory, SAGT 2021 - Virtual, Online Duration: 21 Sep 2021 → 24 Sep 2021 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12885 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Symposium on Algorithmic Game Theory, SAGT 2021 |
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City | Virtual, Online |

Period | 21/09/21 → 24/09/21 |

## Keywords

- Gerrymandering
- Parameterized complexity
- Representative set

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science