Inapproximability of $H$-Transversal/Packing

TL;DR

This paper proves strong inapproximability results for the $H$-Transversal and $H$-Packing problems when $H$ is 2-connected, showing they are nearly as hard as well-known hypergraph problems, with some exceptions based on connectivity.

Contribution

It establishes NP-hardness of approximating $H$-Transversal and $H$-Packing for 2-connected $H$, and explores the impact of connectivity on approximation algorithms.

Findings

NP-hard to approximate within a factor of $ ilde{ ext{O}}(k)$ for 2-connected $H$.

Existence of a 1-connected $H$ with an $O( ext{log }k)$ approximation algorithm.

Implications for Feedback Vertex Set in special cases.

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 2-connected, $H$-Transversal and $H$-Packing are almost as hard to approximate as general $k$-Hypergraph Vertex Cover and $k$-Set Packing, so it is NP-hard to approximate them within a factor of $\Omega (k)$ and $\widetilde \Omega (k)$ respectively. We also show that there is a 1-connected $H$ 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.