Strengthening Landmark Heuristics via Hitting Sets Technical Report: Proofs

Let A = {a1, . . . , an} be a set and F = {F1, . . . , Fm} a family of subsets of A. A subset H ⊆ A has the hitting set property, or is a hitting set, iff H ∩ Fi 6= ∅ for 1 ≤ i ≤ m (i.e., H “hits” each set Fi). If we are given a cost function c : A → N, the cost of H is ∑ a∈H c(a). A hitting set is of minimum cost if its cost is minimal among all hitting sets. The problem of finding a minimum-cost hitting set for family F and cost function c is denoted by 〈F , c〉, and the cost of its solution by min(F , c). A relaxation for 〈F , c〉 is a problem 〈F ′, c′〉 such that c′ ≤ c, and for all F ′ ∈ F ′ there is F ∈ F with F ⊆ F ′. In words, 〈F , c〉 can be relaxed by reducing costs, dropping sets from F , or enlarging elements of F . Determining the existence of a hitting set for a given cost bound is a classic problem in computer science, one of the first problems to be shown NP-complete (Kar72). Lemma 1. If 〈F ′, c′〉 is a relaxation of 〈F , c〉, then min(F ′, c′) ≤ min(F , c). Furthermore, if {〈Fi, ci〉} is a collection of relaxations of F such that ∑ i ci ≤ c, then ∑ i min(Fi, ci) ≤ min(F , c).