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.

中文翻译：

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图上的Gerrymandering：计算复杂度和参数化算法

将一个区域划分为多个区域以青睐特定的候选人或政党通常被称为“嫁妆”。在本文中，我们研究了Cohen-Zemach等人提出的图论设置中的gerrymandering问题。[AAMAS 2018]。我们在本文中的贡献是双重的，在概念上和在计算上。我们首先解决伊藤等人提出的开放性问题。[AAMAS 2019]关于当输入图是路径时问题的计算复杂性。接下来，我们建议对其模型进行推广，其中输入包括代表选民集合的$ n $顶点图，一组$ m $候选人$ \ mathcal {C} $，权重函数$ w_v：\ mathcal {C} \ rightarrow {\ mathbb Z} ^ + $代表每个投票者$ v \ in V（G）$，代表投票者对候选人的偏爱，杰出候选人$ p \ in \ mathcal {C} $ 和一个正整数$ k $。目的是确定是否可以将顶点集划分为$ k $成对的不相交的连接集（区）。st $ p $比任何其他候选者赢得更多的区。即使$ k = 2 $，$ m = 2 $和$ G $是完整的二部图（实际上是$ K_ {2，n} $）或完整图，也知道问题出在NPC上。这意味着在搜索FPT算法时，我们需要专注于参数$ n $或forest的子类。为了避免这些棘手的结果，我们针对在时间为$ 2.619 ^ {k}（n + m）^ {O（1）} $和$ 2 ^ {k}（n + m）的路径上运行的问题给出了确定性和随机算法^ {O（1）} $。此外，我们证明了一般图上的问题在$ 2 ^ n（n + m）^ {O（1）} $的时间内可以解决。