A succinct and approximate greedy algorithm for the Minimum Set Cover Problem

Abstract

The Minimum Set Cover Problem (MSCP) is a combinatorial optimization problem belonging to the NP-Hard class in computer science. For this reason, there is no algorithm that in the worst case ensures finding an optimal solution in polynomial-time. For a given universe $X$, the popular greedy heuristic, called Greedy-SetCover, is the main theoretical contribution to obtain an approximate solution for the MSCP in polynomial-time, offering an optimal approximate ratio of $(ln\left|X\right|+1)$. In this article, we propose an approximate algorithm for MSCP within a succinct representation of the input dataset, whose empirical performance improves Greedy-SetCover both in quality and execution time, while offering the same optimal approximation ratio for the problem. Our experiments show that the proposed algorithm is magnitudes of times faster than the aforementioned greedy one, obtaining on average a cardinality much closer to the optimal solution. Furthermore, because we work on a succinct representation that allows us to compute operations between sets using bitwise operators, we can process much larger datasets than state-of-the-art solutions. As a result, our proposal is also a suitable alternative for processing large datasets as required by the current Big Data era.