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Inapproximability of H-Transversal/Packing

Authors Venkatesan Guruswami, Euiwoong Lee

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Venkatesan Guruswami
Euiwoong Lee

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Venkatesan Guruswami and Euiwoong Lee. Inapproximability of H-Transversal/Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 284-304, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.284

Abstract

Given an undirected graph G=(V,E) and a fixed pattern graph H with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest subset S of vertices such that the subgraph induced by V - S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S1, ..., Sm such that the subgraph induced by each Si 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 Omega(k / polylog(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.
Keywords
  • Constraint Satisfaction Problems
  • Approximation resistance

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