A note on local search for hitting sets
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Let π be a property of pairs (G,Z), where G is a graph and Z⊆ V(G). In the minimum π-hitting set problem, given an input graph G, we want to find a smallest set X⊆ V(G) such that X intersects every set Z⊆ V(G) such that (G,Z) has the property π. An important special case is that π is satisfied by (G,Z) exactly if G[Z] is isomorphic to one of graphs in a finite set ℱ; in this minimum ℱ-hitting set problem, X needs to hit all appearances of the graphs from ℱ as induced subgraphs of G. In this note, we show that the local search argument of Har-Peled and Quanrud gives a PTAS for the minimum ℱ-hitting set problem for graphs from any class with polynomial expansion. Moreover, we argue that the local search argument applies more generally to all properties π such that one can test whether X is a π-hitting set in polynomial time and G[Z] has bounded diameter whenever (G,Z) satisfies π; this is a common generalization of the minimum ℱ-hitting set problem and minimum r-dominating set problem. Finaly, we note that the analogous claim also holds for the dual problem of finding the maximum number of disjoint sets Z such that (G,Z) has the property π; this generalizes maximum F-matching, maximum induced F-matching, and maximum r-independent set problems.
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