操纵选区赢得选举：细粒度的复杂性
Manipulating Districts to Win Elections: Fine-Grained Complexity
摘要：
选区划分是一种操纵地区边界和位置的做法，以实现特定政党的政治优势。Lewenberg，Lev和Rosenschein [AAMAS 2017]发起了基于地理的操纵问题的算法研究，选民必须在离他们最近的投票箱投票。在这种不公正划分选区的变体中，对于给定的一组可能的投票箱位置和$n$选民的已知政治偏好，任务是从$m$可能的位置中识别出$k$箱的位置，以保证某个政党至少在$l$区中获胜。这里，整数$k$和$l$是某个选定的参数。据了解，这个问题是NP-完全已经为4个政党和之前，我们的工作只有启发式算法为这个问题的开发。我们从参数化和细粒度复杂性的角度出发，对选区划分问题进行了深入的研究，并给出了其计算复杂性的渐近匹配的上下界.我们证明了问题是W[1]-hard的$k+n$参数化，它不承认一个$f(n,k)\cdot m^{o(\sqrt{k})}$算法的任何功能$f$的$k$和$n$，除非指数时间假设（ETH）失败。我们的下限已经对$2$政党成立。另一方面，我们给出了一个算法，解决了一个常数的当事人在时间$(m+n)^{O(\sqrt{k})}$的问题。
英文摘要：
Gerrymandering is a practice of manipulating district boundaries and locations in order to achieve a political advantage for a particular party. Lewenberg, Lev, and Rosenschein [AAMAS 2017] initiated the algorithmic study of a geographically-based manipulation problem, where voters must vote at the ballot box closest to them. In this variant of gerrymandering, for a given set of possible locations of ballot boxes and known political preferences of $n$ voters, the task is to identify locations for $k$ boxes out of $m$ possible locations to guarantee victory of a certain party in at least $l$ districts. Here integers $k$ and $l$ are some selected parameter. It is known that the problem is NP-complete already for 4 political parties and prior to our work only heuristic algorithms for this problem were developed. We initiate the rigorous study of the gerrymandering problem from the perspectives of parameterized and fine-grained complexity and provide asymptotically matching lower and upper bounds on its computational complexity. We prove that the problem is W[1]-hard parameterized by $k+n$ and that it does not admit an $f(n,k)\cdot m^{o(\sqrt{k})}$ algorithm for any function $f$ of $k$ and $n$ only, unless Exponential Time Hypothesis (ETH) fails. Our lower bounds hold already for $2$ parties. On the other hand, we give an algorithm that solves the problem for a constant number of parties in time $(m+n)^{O(\sqrt{k})}$.
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