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arXiv · 0906.1557

Uniform unweighted set cover: The power of non-oblivious local search

Asaf Levin·Uri Yovel
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Abstract

We are given n base elements and a finite collection of subsets of them. The size of any subset varies between p to k (p < k). In addition, we assume that the input contains all possible subsets of size p. Our objective is to find a subcollection of minimum-cardinality which covers all the elements. This problem 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 packing phases: at each phase we pick a collection of disjoint subsets covering i new elements, starting from i = k down to i = p+1. At a final step we cover the remaining base elements by the subsets of size p. We derive the exact performance guarantee of this algorithm for all values of k and p, which is less than Hk, where Hk is the k'th harmonic number. However, the algorithm exhibits the known improvement methods over the greedy one for the unweighted k-set cover problem (in which subset sizes are only restricted not to exceed k), and hence it serves as a benchmark for our improved algorithm. The improved algorithm for the special case of (p,k) = (2,4) is based on non-oblivious local search: it starts with a feasible cover, and then repeatedly tries to replace sets of size 3 and 4 so as to maximize an objective function which prefers big sets over small ones. For this case, our generic algorithm achieves an asymptotic approximation ratio of 1.5 + epsilon, and the local search algorithm achieves a better ratio, which is bounded by 1.458333... + epsilon.

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Asaf Levin, Uri Yovel. 2009-06-08. Uniform unweighted set cover: The power of non-oblivious local search. https://arxiv.org/abs/0906.1557

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