SDP Approach to Quadratic Vertex-Disjoint Paths Problem

TL;DR

This paper introduces an SDP-based branch-and-bound method for solving the quadratic vertex-disjoint paths problem, demonstrating superior performance over Gurobi on large instances.

Contribution

It formulates the problem as a binary quadratic program, applies graph reduction, and develops an SDP relaxation with a specialized solver within a branch-and-bound framework.

Findings

The proposed method outperforms Gurobi on large-scale instances.

SDP relaxation provides effective bounds for the branch-and-bound algorithm.

Systematic graph reduction manages problem dimensionality effectively.

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.