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SDP Approach to Quadratic Vertex-Disjoint Paths Problem

Mingming Xu, Hao Hu

Abstract

We study the quadratic $k$-vertex-disjoint paths problem (Q-$k$-VDP), which seeks $k$ vertex-disjoint paths in a directed graph that minimize a nonconvex quadratic objective function. We formulate the problem as a binary quadratic program and apply a systematic graph reduction to manage its dimensionality. To obtain a tractable bounding model, we drop the subtour-elimination constraints and derive a semidefinite programming (SDP) relaxation. We then solve this relaxed model within a branch-and-bound framework, where the bounds are computed from the SDP relaxation using a tailored alternating direction method of multipliers. Computational results show that our proposed method consistently outperforms Gurobi by solving more instances to optimality, especially on challenging large-scale instances.

SDP Approach to Quadratic Vertex-Disjoint Paths Problem

Abstract

We study the quadratic -vertex-disjoint paths problem (Q--VDP), which seeks vertex-disjoint paths in a directed graph that minimize a nonconvex quadratic objective function. We formulate the problem as a binary quadratic program and apply a systematic graph reduction to manage its dimensionality. To obtain a tractable bounding model, we drop the subtour-elimination constraints and derive a semidefinite programming (SDP) relaxation. We then solve this relaxed model within a branch-and-bound framework, where the bounds are computed from the SDP relaxation using a tailored alternating direction method of multipliers. Computational results show that our proposed method consistently outperforms Gurobi by solving more instances to optimality, especially on challenging large-scale instances.

Paper Structure

This paper contains 23 sections, 10 theorems, 34 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Graphs
  4. Semidefinite Programming
  5. Binary Quadratic Program
  6. Disjoint Union Graph Reduction
  7. Fixed-Zero Arcs
  8. Fixed-One Arcs
  9. Graph Reduction
  10. SDP Relaxation of the Subtour-Relaxed Model
  11. SDP Relaxation and Facial Reduction
  12. Slater's Condition
  13. Dimensionality Reduction and Bound Analysis
  14. Slater's condition on the Reduced Graph
  15. The Alternating Direction Method of Multipliers
  16. ...and 8 more sections

Key Result

Theorem 2.1

Either (P) is strictly feasible, or the auxiliary system $0 \neq \mathbf{A}^*(y) \succeq 0$ with $\langle b,y\rangle \le 0$ is feasible. If (P) is feasible, this auxiliary system requires $\langle b,y\rangle = 0$. In this case, the primal feasible set is contained within the proper face: The vector $\mathbf{A}^*(y)$ is then called an exposing vector of $\mathcal{F}$. $\blacktriangleleft$$\blacktr

Figures (3)

  • Figure 1: Directed planar graph $\mathcal{G}$ with source--target pairs $(1, 2)$ and $(3, 4)$ as described in Example \ref{['ex:slater_failure']}. This instance, consisting of six nodes and eight arcs, is used to illustrate the failure of Slater's condition in the SDP relaxation.
  • Figure 2: Disjoint union graph $\mathcal{G}^{1}\sqcup\mathcal{G}^{2}$, constructed from two isomorphic copies of the base graph $\mathcal{G}$ in Figure \ref{['fig:example_5_1']}.
  • Figure 3: Reduced graph $G$ after removing fixed arcs in the disjoint union graph in Figure \ref{['fig:disjoint_union']}.

Theorems & Definitions (19)

  • Theorem 2.1: Theorem of the alternative
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 9 more