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

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. 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.