Structural Segmentation of the Minimum Set Cover Problem: Exploiting Universe Decomposability for Metaheuristic Optimization

TL;DR

This paper introduces a structural decomposition approach for the Minimum Set Cover Problem, leveraging universe segmentability to improve heuristic optimization efficiency and solution quality.

Contribution

It proposes a novel preprocessing method using union-find to detect universe segments, enabling independent subproblem solving with metaheuristics.

Findings

Exploiting universe segmentability improves solution quality.

Decomposition enhances scalability for large instances.

Efficient bit-level set representation supports practical implementation.

Abstract

The Minimum Set Cover Problem (MSCP) is a classical NP-hard combinatorial optimization problem with numerous applications in science and engineering. Although a wide range of exact, approximate, and metaheuristic approaches have been proposed, most methods implicitly treat MSCP instances as monolithic, overlooking potential intrinsic structural properties of the universe. In this work, we investigate the concept of \emph{universe segmentability} in the MSCP and analyze how intrinsic structural decomposition (universe segmentability) can be exploited to enhance heuristic optimization. We propose an efficient preprocessing strategy based on disjoint-set union (union--find) to detect connected components induced by element co-occurrence within subsets, enabling the decomposition of the original instance into independent subproblems. Each subproblem is solved using the GRASP metaheuristic, and partial solutions are combined without compromising feasibility. Extensive experiments on standard benchmark instances and large-scale synthetic datasets show that exploiting natural universe segmentation consistently improves solution quality and scalability, particularly for large and structurally decomposable instances. These gains are supported by a succinct bit-level set representation that enables efficient set operations, making the proposed approach computationally practical at scale.