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Representability of forests via generalized subtour elimination constraints

Matheus J. Ota

TL;DR

The paper addresses which forests can be represented as integer points inside a GSEC-based polytope $P(f;G)$ by introducing an edge-consistent framework and a complete characterization via the $\\Phi(f)$ construction, tying representability to downward-closedness and the minimal infeasibility property. It analyzes how to choose RHS functions $f$ to obtain weakest or strongest relaxations and extends the theory to include additional constraints through a projection polyhedron $\\mathcal{Q}$, enabling broader VRP extensions. The authors apply the theory to recover and extend VRP formulations, including a bike sharing rebalancing formulation and a robust capacitated minimum spanning tree (RCMST) formulation, demonstrating the practical impact of GSEC-based modeling beyond traditional permutation-invariant settings. Overall, the work broadens the applicability of generalized subtour elimination constraints to a wider class of network design and vehicle routing problems, providing exact representability conditions and constructive RHS design for both standard and robust optimization contexts.

Abstract

Generalized subtour elimination constraints (GSECs) are widely used in state-of-the-art exact algorithms for vehicle routing and network design problems, as their right-hand sides often capture problem-specific feasibility conditions of each solution component. In this work, we present the first characterization of the families of forests that can be represented as the integer points inside a polytope defined by GSECs. This result generalizes a recent framework developed for vehicle routing problems under uncertainty and broadens the applicability of GSEC-based formulations to a wider class of combinatorial problems. In particular, using our characterization, we recover vehicle routing formulations that could not be obtained with previous results. Additionally, we show that GSECs can naturally model a robust variant of the capacitated minimum spanning tree problem.

Representability of forests via generalized subtour elimination constraints

TL;DR

The paper addresses which forests can be represented as integer points inside a GSEC-based polytope by introducing an edge-consistent framework and a complete characterization via the construction, tying representability to downward-closedness and the minimal infeasibility property. It analyzes how to choose RHS functions to obtain weakest or strongest relaxations and extends the theory to include additional constraints through a projection polyhedron , enabling broader VRP extensions. The authors apply the theory to recover and extend VRP formulations, including a bike sharing rebalancing formulation and a robust capacitated minimum spanning tree (RCMST) formulation, demonstrating the practical impact of GSEC-based modeling beyond traditional permutation-invariant settings. Overall, the work broadens the applicability of generalized subtour elimination constraints to a wider class of network design and vehicle routing problems, providing exact representability conditions and constructive RHS design for both standard and robust optimization contexts.

Abstract

Generalized subtour elimination constraints (GSECs) are widely used in state-of-the-art exact algorithms for vehicle routing and network design problems, as their right-hand sides often capture problem-specific feasibility conditions of each solution component. In this work, we present the first characterization of the families of forests that can be represented as the integer points inside a polytope defined by GSECs. This result generalizes a recent framework developed for vehicle routing problems under uncertainty and broadens the applicability of GSEC-based formulations to a wider class of combinatorial problems. In particular, using our characterization, we recover vehicle routing formulations that could not be obtained with previous results. Additionally, we show that GSECs can naturally model a robust variant of the capacitated minimum spanning tree problem.

Paper Structure

This paper contains 11 sections, 13 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Generalized subtour elimination constraints and edge-based formulations for vehicle routing problems
  3. Representable families of forests
  4. The main characterization
  5. Different RHS functions
  6. Extension and the case of vehicle routing problems
  7. Vehicle routing problems and componentwise feasibility.
  8. Subadditive functions and problem applications
  9. Bike sharing rebalancing problem
  10. Robust capacitated minimum spanning tree problem
  11. Concluding remarks

Key Result

Lemma 1

If $\mathcal{F}$ is an edge-consistent representable family of forests, then $\mathcal{F}$ has the minimal infeasibility property.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Lemma 1
  • ...and 40 more