arXiv 2102.098892021-02cs.DScs.GT

图上的Gerrymandering：计算复杂性和参数化算法

Gerrymandering on graphs: Computational complexity and parameterized algorithms

Sushmita Gupta, Pallavi Jain, Fahad Panolan, Sanjukta Roy, Saket Saurabh

摘要：

将一个地区划分为有利于特定候选人或政党的地区通常被称为gerrymandering。在本文中，我们研究了Cohen-Zemach等人[AAMAS 2018]提出的图论环境中的gerrymandering问题。我们在这篇文章中的贡献是双重的，概念和计算。我们首先解决了Ito等人提出的公开问题。[AAMAS 2019]关于当输入图是路径时问题的计算复杂性。接下来，我们提出了他们模型的推广，其中输入由代表选民集的$n$顶点上的图，$m$候选者集$\mathcal{C}$，用于每个投票者的权重函数$w_v: \mathcal{C}\rightarrow {\mathbb Z}^+$，其表示投票者对候选者的偏好$v\in V(G)$、区分候选者$p\in \mathcal{C}$和正整数$k$。目标是决定是否可以将顶点集划分成$k$成对不相交的连通集（区），S.T $p$比任何其他候选者赢得更多的区。即使$k=2$、$m=2$和$G$是完全二分图（实际上是$K_{2,n}$）或完全图，该问题也被称为NPC。这意味着在搜索FPT算法时，我们需要关注参数$n$或森林的子类。避开这些棘手的结果，我们给出了一个确定性和一个随机算法的时间运行路径$2.619^{k}(n+m)^{O(1)}$和$2^{k}(n+m)^{O(1)}$，分别。此外，我们证明了一般图上的问题是可解的时间$2^n (n+m)^{O(1)}$。我们的算法结果使用复杂的技术工具，如代表集家庭和快速傅立叶变换为基础的多项式乘法，其（可能是第一次）的应用所产生的问题，在社会选择理论和/或博弈论可能是独立的利益社会。

英文摘要：

Partitioning a region into districts to favor a particular candidate or a party is commonly known as gerrymandering. In this paper, we investigate the gerrymandering problem in graph theoretic setting as proposed by Cohen-Zemach et al. [AAMAS 2018]. Our contributions in this article are two-fold, conceptual and computational. We first resolve the open question posed by Ito et al. [AAMAS 2019] about the computational complexity of the problem when the input graph is a path. Next, we propose a generalization of their model, where the input consists of a graph on $n$ vertices representing the set of voters, a set of $m$ candidates $\mathcal{C}$, a weight function $w_v: \mathcal{C}\rightarrow {\mathbb Z}^+$ for each voter $v\in V(G)$ representing the preference of the voter over the candidates, a distinguished candidate $p\in \mathcal{C}$, and a positive integer $k$. The objective is to decide if one can partition the vertex set into $k$ pairwise disjoint connected sets (districts) s.t $p$ wins more districts than any other candidate. The problem is known to be NPC even if $k=2$, $m=2$, and $G$ is either a complete bipartite graph (in fact $K_{2,n}$) or a complete graph. This means that in search for FPT algorithms we need to either focus on the parameter $n$, or subclasses of forest. Circumventing these intractable results, we give a deterministic and a randomized algorithms for the problem on paths running in times $2.619^{k}(n+m)^{O(1)}$ and $2^{k}(n+m)^{O(1)}$, respectively. Additionally, we prove that the problem on general graphs is solvable in time $2^n (n+m)^{O(1)}$. 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 game theory may be of independent interest to the community.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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