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 $S_1, \ldots, S_m \subseteq V_G$ 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 m...
Highlights
Given a collection of subsets S1, ..., Sm of the underlying set U, the Set Transversal problem asks to find the smallest subset of U that intersects every Si, and the Set Packing problem asks to find the largest subcollection Si1, ..., Sim which are pairwise disjoint
Set Transversal and Set Packing are known as k-Hypergraph
We give a simple sufficient condition that implies strong inapproximability – if H is 2-vertex connected, H-Transversal and H-Packing are almost as hard to approximate as k-HypergraphVertex cover (k-HVC) and k-Set Packing (k-SP)
Summary
Given a collection of subsets S1, ..., Sm of the underlying set U , the Set Transversal problem asks to find the smallest subset of U that intersects every Si, and the Set Packing problem asks to find the largest subcollection Si1 , ..., Sim which are pairwise disjoint. It is clear that optimum of the former is always at least that of the latter (i.e. weak duality holds). Given a large graph G = (VG, EG) and a fixed graph H = (VH , EH ) with k vertices, one of the natural attempts to further restrict set systems is to set U = VG, and take the collection of subsets to be all copies of H in G (formally defined in the subsection) This natural representation in graphs often results in a deeper understanding of the underlying structure and better algorithms, with Maximum Matching (H = K2) being the most well-known example. We give a simple sufficient condition that implies strong inapproximability – if H is 2-vertex connected, H-Transversal and H-Packing are almost as hard to approximate as k-HVC and k-SP. We state our main results, review related work, and state potential future directions
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