Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Set System Approximation for Binary Integer Programs: Reformulations and Applications

Ningji Wei

TL;DR

This work develops a unified set-system view of binary integer programs, showing that covering and elimination inequalities correspond to tight inner and outer monotone approximations of the binary solution space. It builds a comprehensive toolkit—defining monotone operators, cut-cocut algebra, and precise reformulations—that extends classical set-cover polyhedral results to general BIPs and recovers well-known structure correspondences (e.g., paths vs. cuts, cycles vs. dominating sets). It further introduces latent monotone reformulation techniques, including bilinear linearization without auxiliary variables, bimonotone cuts, and interval-system decompositions, enabling exact reformulations for nonlinear and latent monotone problems. A distributionally robust network site selection case study demonstrates the framework’s flexibility and computational benefits, showing how different monotone subsystems yield hybrid implementations with practical performance gains. Overall, the paper provides a rigorous, generalizable approach to inner/outer approximation, polyhedral analysis, and reformulation for nonlinear BIPs with broad applicability.

Abstract

Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these inequalities as primitives for set system approximation in binary integer programs (BIPs). We show that arbitrary set systems admit tight inner and outer monotone approximations, exactly corresponding to covering and elimination inequalities. Building on this, we develop a toolkit that both recovers classical structural correspondences (e.g., paths vs. cuts, spanning trees vs. cycles) and extends polyhedral tools from set covering to general BIPs, including facet conditions and lifting methods. We also propose new reformulation techniques for nonlinear and latent monotone systems, such as auxiliary-variable-free bilinear linearization, bimonotone cuts, and interval decompositions. A case study on distributionally robust network site selection illustrates the framework's flexibility and computational benefits. Overall, this unified view clarifies inner/outer approximation criteria, extends classical polyhedral analysis, and provides broadly applicable reformulation strategies for nonlinear BIPs.

Set System Approximation for Binary Integer Programs: Reformulations and Applications

TL;DR

This work develops a unified set-system view of binary integer programs, showing that covering and elimination inequalities correspond to tight inner and outer monotone approximations of the binary solution space. It builds a comprehensive toolkit—defining monotone operators, cut-cocut algebra, and precise reformulations—that extends classical set-cover polyhedral results to general BIPs and recovers well-known structure correspondences (e.g., paths vs. cuts, cycles vs. dominating sets). It further introduces latent monotone reformulation techniques, including bilinear linearization without auxiliary variables, bimonotone cuts, and interval-system decompositions, enabling exact reformulations for nonlinear and latent monotone problems. A distributionally robust network site selection case study demonstrates the framework’s flexibility and computational benefits, showing how different monotone subsystems yield hybrid implementations with practical performance gains. Overall, the paper provides a rigorous, generalizable approach to inner/outer approximation, polyhedral analysis, and reformulation for nonlinear BIPs with broad applicability.

Abstract

Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these inequalities as primitives for set system approximation in binary integer programs (BIPs). We show that arbitrary set systems admit tight inner and outer monotone approximations, exactly corresponding to covering and elimination inequalities. Building on this, we develop a toolkit that both recovers classical structural correspondences (e.g., paths vs. cuts, spanning trees vs. cycles) and extends polyhedral tools from set covering to general BIPs, including facet conditions and lifting methods. We also propose new reformulation techniques for nonlinear and latent monotone systems, such as auxiliary-variable-free bilinear linearization, bimonotone cuts, and interval decompositions. A case study on distributionally robust network site selection illustrates the framework's flexibility and computational benefits. Overall, this unified view clarifies inner/outer approximation criteria, extends classical polyhedral analysis, and provides broadly applicable reformulation strategies for nonlinear BIPs.

Paper Structure

This paper contains 19 sections, 30 theorems, 66 equations, 1 figure, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Set System Approximation
  5. Tightest Monotone Approximations of Set Systems
  6. Approximation-Based Reformulations for General BIPs
  7. Combinatorial and Polyhedral Structures in BIPs
  8. Recovering Structural Correspondence in Classic Problems
  9. Generalizing Polyhedral Analysis for BIPs
  10. Reformulations via Latent Monotone Systems
  11. Bimonotone Cuts
  12. Interval System Decomposition
  13. Case Study: Network Site Selection under Uncertainty
  14. Problem Setting
  15. Performance Comparison of Four Implementations
  16. ...and 4 more sections

Key Result

Theorem 1

For every set system $\Omega$, we have where $\mathcal{C}(\widehat{\overline\Omega})$ and $\mathcal{C}(\widehat{\overline{\uparrow\Omega}})$ are the tightest inner and outer approximations of $\Omega$ using upper-systems, respectively. Moreover, equality holds throughout if and only if $\Omega$ is upper-closed.

Figures (1)

  • Figure 1: Two set systems, $\Omega_1$ (left) and $\Omega_2$ (right), each defined over the ground set $\Delta=\{1,2,3,4\}$. Each curve encloses all subsets belonging to $\Omega_1$ (left) or $\Omega_2$ (right).

Theorems & Definitions (65)

  • Definition 1: Monotone Systems
  • Definition 2
  • Theorem 1: Upper Approximation
  • Corollary 1: Lower Approximation
  • Corollary 2
  • Example 1
  • Lemma 1
  • Theorem 2
  • Corollary 3
  • Remark 1
  • ...and 55 more