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

Isidora Hernández, Héctor Ferrada, Cristóbal A. Navarro

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.

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

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.

Paper Structure

This paper contains 27 sections, 1 theorem, 8 equations, 10 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Structural Segmentation of the MSCP
  4. Universe Segmentability
  5. Feasibility Preservation
  6. Union--Find-Based Decomposition
  7. Integration with GRASP
  8. Construction Phase and Evaluation Functions
  9. Computational Complexity of GRASP
  10. Preprocessing phase.
  11. Randomized construction phase.
  12. GRASP iterations.
  13. Universe Segmentation and Independent Subinstances
  14. Parallel GRASP with Universe Segmentation
  15. Computational Complexity of GRASP-SU
  16. ...and 12 more sections

Key Result

Proposition 1

Let $(\mathcal{X}, \mathcal{F})$ be an MSCP instance whose co-occurrence graph decomposes into connected components $\mathcal{X}_1, \ldots, \mathcal{X}_k$. For each $i$, let $\mathcal{C}_i \subseteq \mathcal{F}_i$ be a feasible cover of $\mathcal{X}_i$. Then $\mathcal{C} = \bigcup_{i=1}^k \mathcal{C

Figures (10)

  • Figure 1: Illustration of an instance $(\mathcal{X}, \mathcal{F})$ of the MSCP, with $|\mathcal{X}|=12$ elements and $\mathcal{F}=\{S_1,\ldots,S_7\}$, where $S_1 = \{1, 2, 5, 6, 9\}, S_2 = \{3, 4, 7, 8\}, S_3 = \{2, 3, 4\}, S_4 = \{2, 3, 6, 7, 9, 10\}, S_5 = \{4, 8, 11\}, S_6 = \{9, 10, 11, 12\}, S_7 = \{1, 5\}$. The optimal solution for this example is the minimum set-covering $\mathcal{C^*}=\{S_1,S_2,S_6\}$.
  • Figure 2: Illustrative example of a MSCP instance. (Left) Family of subsets $\mathcal{F}$ defined over the universe $\mathcal{X}$. (Right) Co-occurrence graph induced by $\mathcal{F}$, where an edge connects two elements if they appear together in at least one subset. The graph decomposes into three connected components, motivating universe segmentation into independent subinstances.
  • Figure 3: Relative percentage deviation (RPD) obtained by Greedy and GRASP on OR-Library MSCP instances.
  • Figure 4: Relative percentage deviation for railway instances, comparing Greedy and GRASP.
  • Figure 5: Solution cardinality and execution time of GRASP-UF and PAR-GRASP, both executed with 32 CPU cores, on randomly generated MSCP instances of size $(n,m) = (10{,}000, 20{,}000)$. The number of universe groups produced by segmentation varies in $\{1,2,4,8,16,32,64\}$. The metric RPD$^\ast$ denotes the relative percentage deviation with respect to the solution cardinality obtained by the Greedy algorithm.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Definition 1: Universe Segmentability
  • Proposition 1
  • proof