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Benders decomposition for the large-scale probabilistic set covering problem

Jie Liang, Cheng-Yang Yu, Wei Lv, Wei-Kun Chen, Yu-Hong Dai

TL;DR

This work tackles the large-scale probabilistic set covering problem (PSCP) where each constraint row is a random 0-1 vector with a finite discrete distribution and each row i must be satisfied with probability at least 1−ε_i. It introduces a Benders decomposition (BD) that keeps the master problem size at the number of columns n, independent of the number of scenarios s, and provides a polynomial-time separation for Benders cuts, making it well-suited for large PSCP instances. The authors derive a multi-cut BD formulation from the big-$M$ MIP, and develop three enhancements—tight initial cuts, a relaxation-enforced neighborhood search (RENS) heuristic, and mixed integer rounding (MIR)–enhanced Benders cuts—to strengthen the formulation and accelerate convergence. Numerical results show that the BD approach significantly outperforms state-of-the-art MIP solvers on both small and large instances, including problems with up to 500 rows, 5000 columns, and 2000 scenarios, underscoring its practical impact for large-scale, scenario-based covering problems.

Abstract

In this paper, we consider a probabilistic set covering problem (PSCP) in which each 0-1 row of the constraint matrix is random with a finite discrete distribution, and the objective is to minimize the total cost of the selected columns such that each row is covered with a prespecified probability. We develop an effective decomposition algorithm for the PSCP based on the Benders reformulation of a standard mixed integer programming (MIP) formulation. The proposed Benders decomposition (BD) algorithm enjoys two key advantages: (i) the number of variables in the underlying Benders reformulation is equal to the number of columns but independent of the number of scenarios of the random data; and (ii) the Benders feasibility cuts can be separated by an efficient polynomial-time algorithm, which makes it particularly suitable for solving large-scale PSCPs. We enhance the BD algorithm by using initial cuts to strengthen the relaxed master problem, implementing an effective heuristic procedure to find high-quality feasible solutions, and adding mixed integer rounding enhanced Benders feasibility cuts to tighten the problem formulation. Numerical results demonstrate the efficiency of the proposed BD algorithm over a state-of-the-art MIP solver. Moreover, the proposed BD algorithm can efficiently identify optimal solutions for instances with up to 500 rows, 5000 columns, and 2000 scenarios of the random rows.

Benders decomposition for the large-scale probabilistic set covering problem

TL;DR

This work tackles the large-scale probabilistic set covering problem (PSCP) where each constraint row is a random 0-1 vector with a finite discrete distribution and each row i must be satisfied with probability at least 1−ε_i. It introduces a Benders decomposition (BD) that keeps the master problem size at the number of columns n, independent of the number of scenarios s, and provides a polynomial-time separation for Benders cuts, making it well-suited for large PSCP instances. The authors derive a multi-cut BD formulation from the big- MIP, and develop three enhancements—tight initial cuts, a relaxation-enforced neighborhood search (RENS) heuristic, and mixed integer rounding (MIR)–enhanced Benders cuts—to strengthen the formulation and accelerate convergence. Numerical results show that the BD approach significantly outperforms state-of-the-art MIP solvers on both small and large instances, including problems with up to 500 rows, 5000 columns, and 2000 scenarios, underscoring its practical impact for large-scale, scenario-based covering problems.

Abstract

In this paper, we consider a probabilistic set covering problem (PSCP) in which each 0-1 row of the constraint matrix is random with a finite discrete distribution, and the objective is to minimize the total cost of the selected columns such that each row is covered with a prespecified probability. We develop an effective decomposition algorithm for the PSCP based on the Benders reformulation of a standard mixed integer programming (MIP) formulation. The proposed Benders decomposition (BD) algorithm enjoys two key advantages: (i) the number of variables in the underlying Benders reformulation is equal to the number of columns but independent of the number of scenarios of the random data; and (ii) the Benders feasibility cuts can be separated by an efficient polynomial-time algorithm, which makes it particularly suitable for solving large-scale PSCPs. We enhance the BD algorithm by using initial cuts to strengthen the relaxed master problem, implementing an effective heuristic procedure to find high-quality feasible solutions, and adding mixed integer rounding enhanced Benders feasibility cuts to tighten the problem formulation. Numerical results demonstrate the efficiency of the proposed BD algorithm over a state-of-the-art MIP solver. Moreover, the proposed BD algorithm can efficiently identify optimal solutions for instances with up to 500 rows, 5000 columns, and 2000 scenarios of the random rows.
Paper Structure (19 sections, 5 theorems, 26 equations, 5 figures)

This paper contains 19 sections, 5 theorems, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Contributions and Outlines
  3. Literature Review
  4. Mixed integer programming formulation
  5. Benders decomposition
  6. Benders reformulation
  7. Implementation
  8. Improving performance of the Benders decomposition
  9. Initialization of relaxed master problem
  10. A customized relaxation enforced neighborhood search procedure
  11. MIR-enhanced Benders feasibility cuts
  12. Mixed integer rounding
  13. Comparison with the strengthened Benders feasibility cut in Cordeau2019
  14. Numerical results
  15. Testsets
  16. ...and 4 more sections

Key Result

Proposition 3

Let $x^\text{LP}\xspace$ be a feasible solution of the LP relaxation of problem mp-ipscp, and $\mathcal{N}^0$ and $\mathcal{N}^1$ be defined as in N01. Then subproblem mp-ipscp-sp has a feasible solution.

Figures (5)

  • Figure 1: Performance profiles of the CPU times and end gaps on the instances in testset $\text{T}_1$.
  • Figure 2: Performance profiles of the CPU times and end gaps on the instances in testset $\text{T}_2$. (a) and (b): grouped by $\epsilon$; (c) and (d): grouped by $s$; (e) and (f): grouped by independent and correlated instances.
  • Figure 3: (a) Performance profiles of the root gaps on the instances in testset $\text{T}_2$; (b) Performance profiles of the primal gaps on the instances with $\epsilon=0.025$ in testset $\text{T}_2$.
  • Figure 4: Performance profiles of the CPU times, end gaps, root gaps, and number of explored nodes on the instances in testset $\text{T}_2$.
  • Figure 5: Performance profile of the primal gaps of the solutions returned by RENS on the instances in testset $\text{T}_2$.

Theorems & Definitions (10)

  • Remark 1
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • Proposition 7
  • proof
  • Proposition 8
  • proof