Uniform unweighted set cover: The power of non-oblivious local search
Asaf Levin, Uri Yovel

TL;DR
This paper introduces approximation algorithms for the uniform unweighted set cover problem, including a greedy approach with performance guarantees and an improved non-oblivious local search method for specific parameters, achieving near-optimal ratios.
Contribution
It provides the first performance guarantees for a greedy algorithm in this setting and develops an improved local search algorithm with better approximation ratios for the (2,4) case.
Findings
The greedy algorithm's performance guarantee is less than Hk for all k and p.
The local search algorithm achieves an approximation ratio of approximately 1.4583 for the (2,4) case.
The algorithms serve as benchmarks for the unweighted k-set cover problem.
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.…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Optimization and Packing Problems · Complexity and Algorithms in Graphs
