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Inapproximability of \(H\)-transversal/packing. (English) Zbl 1371.68099
Summary: 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 \(S_1,\dots, S_m \subseteq V_G\) 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.
Reviewer: Reviewer (Berlin)

68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
68W25 Approximation algorithms
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