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Approximation Algorithm for the Partial Set Multi-Cover Problem
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 , a collection
of sets , a total covering ratio which is a
constant between 0 and 1, each set is associated with a cost
, each element is associated with a covering requirement ,
the goal is to find a minimum cost sub-collection to fully cover at least elements, where element is fully covered
if it belongs to at least sets of . Denote by
the maximum covering requirement. We
present an -bicriteria
approximation algorithm, that is, the output of our algorithm has cost at most
times of the optimal value while the
number of fully covered elements is at least