发表机构
Grainger College of Engineering, Univ. of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷恩格工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对网络非均匀边故障问题，提出k层灵活图连通性模型，为其三个变体分别设计近似算法，其中最小基数变体的近似因子仅依赖q₁、qₖ，多图变体为2-近似。
AI 中文摘要
受网络设计中非均匀边故障的启发，我们引入了一种多层级灵活图连通性模型。在k层灵活图连通性（k-tier FGC）中，输入为带非负边代价的无向图G=(V, E)，以及将边划分为嵌套层级T₁⊆T₂⊆…⊆Tₖ=E的分类，还有非负整数层级需求q₁≤q₂≤…≤qₖ。非空真顶点子集R是安全的，当且仅当它沿某一层级是安全的，即存在i∈[k]使得|δ(R)∩Tᵢ|≥qᵢ。目标是找到代价最小的边子集F⊆E，使得子图(V, F)不存在不安全的割。当k=1时，该问题对应最小代价p边连通生成子图问题，属于APX难问题。我们针对k-tier FGC的三个变体，为每个固定常数k设计了近似算法：(i) 针对k-tier FGC，设计了基于线性规划（LP）的对数近似算法；(ii) 针对最小基数k-tier FGC，设计了组合近似算法，其近似因子仅依赖于层级需求q₁和qₖ；(iii) 针对k层灵活多图连通性（允许使用每条边的多个副本，且每个选中的边副本需支付该边的代价），设计了基于LP的2-近似算法。
英文摘要
Motivated by non-uniform edge failures in network design, we introduce a multi-tier model of flexible graph connectivity. In k-tier Flexible Graph Connectivity (k-tier FGC), the input is an undirected graph G=(V, E) with non-negative edge costs, along with a classification of the edges into nested tiers T_1 subseteq T_2 subseteq ... subseteq T_k = E and non-negative integral tier requirements q_1 <= q_2 <= ... <= q_k. A non-empty proper subset R of vertices is safe if it is safe along one of the tiers, i.e., there exists i in [k] such that |delta(R) cap T_i| >= q_i. The goal is to find a minimum cost subset F subseteq E of edges such that the subgraph (V, F) has no unsafe cuts. The case of k=1 corresponds to the min-cost p-edge-connected spanning subgraph problem which is APX-hard. We design approximation algorithms for every fixed constant k for three variants of k-tier FGC: (i) for k-tier FGC, we design an LP-based logarithmic approximation, (ii) for min-cardinality k-tier FGC, we design a combinatorial approximation whose factor depends only on the tier requirements q_1 and q_k, and (iii) for k-tier Flexible Multi-Graph Connectivity, where we are allowed to use multiple copies of each edge while paying the cost of the edge for each chosen copy of the edge, we design an LP-based 2-approximation.
