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Abstract: We are given n base elements and a finite collection of subsets of them. Thesize of any subset varies between p to k p < k. In addition, we assume thatthe input contains all possible subsets of size p. Our objective is to find asubcollection of minimum-cardinality which covers all the elements. Thisproblem is known to be NP-hard. We provide two approximation algorithms for it,one for the generic case, and an improved one for the special case of p,k =2,4. The algorithm for the generic case is a greedy one, based on packingphases: at each phase we pick a collection of disjoint subsets covering i newelements, starting from i = k down to i = p+1. At a final step we cover theremaining base elements by the subsets of size p. We derive the exactperformance guarantee of this algorithm for all values of k and p, which isless than Hk, where Hk is the k-th harmonic number. However, the algorithmexhibits the known improvement methods over the greedy one for the unweightedk-set cover problem in which subset sizes are only restricted not to exceedk, and hence it serves as a benchmark for our improved algorithm. The improvedalgorithm for the special case of p,k = 2,4 is based on non-oblivious localsearch: it starts with a feasible cover, and then repeatedly tries to replacesets of size 3 and 4 so as to maximize an objective function which prefers bigsets over small ones. For this case, our generic algorithm achieves anasymptotic approximation ratio of 1.5 + epsilon, and the local search algorithmachieves a better ratio, which is bounded by 1.458333

. + epsilon.



Author: Asaf Levin, Uri Yovel

Source: https://arxiv.org/







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