发表机构
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对$(p,q)$-弹性图连通性问题，提出$(p+q)^{O(pq)}$近似算法，通过增广问题和结构结果实现固定参数下的常数近似。
AI 中文摘要
在 $(p,q)$-弹性图连通性问题中，输入包含非负整数 $p$ 和 $q$，以及一个图 $G=(V, E)$，其边被分类为安全边和不安全边，并带有非负边成本。若顶点集的每个非空真子集要么至少有 $p$ 条安全边穿过，要么至少有 $p+q$ 条总边穿过，则图 $G$ 的子图 $H$ 是 $(p,q)$-弹性连通的。目标是找到边集 $F\subseteq E$ 的最小成本子集，使得子图 $(V, F)$ 是 $(p,q)$-弹性连通的。我们给出了该问题的 $(p+q)^{O(pq)}$ 近似算法，这特别意味着对于每个固定的常数 $p$ 和 $q$，存在常数近似。我们通过设计一个 $(p+q)^{O(pq)}$ 近似的增广问题来实现这一点，该增广问题是找到要添加的最小成本边子集，使一个 $(p,q-1)$-弹性连通图变为 $(p,q)$-弹性连通图。该增广算法的基础是一个结构结果，表明所有缺陷割都可以由 $(p+q)^{pq}$ 大小的有向图集合中的最小根割表示。该结构结果由 ChatGPT Astra 发现。
英文摘要
In the $(p,q)$-Flexible Graph Connectivity problem, the input consists of non-negative integers $p$ and $q$ and a graph $G=(V, E)$ whose edges are classified into safe and unsafe edges with non-negative edge costs. A subgraph H of G is $(p,q)$-Flex-Connected if every non-empty proper subset of vertices has either at least $p$ safe edges or at least $p+q$ total edges crossing it. The goal is to find a minimum cost subset $F\subseteq E$ of edges such that the subgraph $(V, F)$ is $(p,q)$-Flex-Connected. We give a $(p+q)^{O(pq)}$-approximation for this problem, which in particular implies a constant approximation for every fixed constants $p$ and $q$. We achieve this by designing a $(p+q)^{O(pq)}$-approximation for the augmentation problem of finding a minimum cost subset of edges to add to make a (p,q-1)-Flex-Connected graph into a (p,q)-Flex-Connected graph. Underlying the augmentation algorithm is a structural result showing that all deficient cuts can be represented by min rooted-cuts in a $(p+q)^{pq}$-sized collection of digraphs. This structural result was discovered by ChatGPT Astra.
