Inapproximability of $H$Transversal/Packing
Abstract
Given an undirected graph $G = (V_G, E_G)$ and a fixed "pattern" graph $H = (V_H, E_H)$ with $k$ vertices, we consider the $H$Transversal and $H$Packing problems. The former asks to find the smallest $S \subseteq V_G$ such that the subgraph induced by $V_G \setminus S$ does not have $H$ as a subgraph, and the latter asks to find the maximum number of pairwise disjoint $k$subsets $S_1, ..., S_m \subseteq V_G$ such that the subgraph induced by each $S_i$ has $H$ as a subgraph. We prove that if $H$ is 2connected, $H$Transversal and $H$Packing are almost as hard to approximate as general $k$Hypergraph Vertex Cover and $k$Set Packing, so it is NPhard to approximate them within a factor of $\Omega (k)$ and $\widetilde \Omega (k)$ respectively. We also show that there is a 1connected $H$ where $H$Transversal admits an $O(\log k)$approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of $H$Transversal where $H$ is a (family of) cycles, we mention the implication of our result to the related Feedback Vertex Set problem, and give a different hardness proof for directed graphs.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 DOI:
 10.48550/arXiv.1506.06302
 arXiv:
 arXiv:1506.06302
 Bibcode:
 2015arXiv150606302G
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 31 pages, 2 figures