Worst Case Analysis of Approximation Algorithm of Abrams et al. for the Set k-Cover Problem

摘要

In this paper, we consider the problem of partitioning a given collection of node sets into k collections such that the average size of collections is the largest, where the size of a collection is defined as the cardinarity of the union of the subsets contained in the collection. More concretely, we give an upper bound on the performance ratio of an approximation algorithm proposed by Abrams et al., which is known to have a performance ratio of at least 1 - 1/e similar or equal to 0.6321 where e is Napier's constant. The proposed upper bound is 1 - (2 - (d+1)root 2)(d+1)/2 for any d = 1 provided that k = o(n) which approaches to 0.75 as d increases.