Approximation Algorithm for the Partial Set Multi-Cover Problem
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Partial set cover problem and set multi-cover problem are two generalizations of set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets S⊆ 2^E, a total covering ratio q which is a constant between 0 and 1, each set S∈ S is associated with a cost c_S, each element e∈ E is associated with a covering requirement r_e, the goal is to find a minimum cost sub-collection S'⊆ S to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least r_e sets of S'. Denote by r_={r_e e∈ E} the maximum covering requirement. We present an (O(r_^2n/ε),1-ε)-bicriteria approximation algorithm, that is, the output of our algorithm has cost at most O(r_^2 n/ε) times of the optimal value while the number of fully covered elements is at least (1-ε)q|E|.
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