On Solving the Quadratic Shortest Path Problem

TL;DR

This paper introduces semidefinite programming relaxations and a branch-and-bound algorithm for the quadratic shortest path problem, achieving strong bounds and solving large instances efficiently.

Contribution

It develops novel SDP relaxations and an effective branch-and-bound method, significantly advancing solution capabilities for the quadratic shortest path problem.

Findings

Semidefinite relaxations provide the strongest bounds for the problem.

The branch-and-bound algorithm solves instances with up to 1300 arcs within an hour.

Numerical results demonstrate the effectiveness of the proposed methods.

Abstract

The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order $m+1$, where $m$ is the number of arcs in the graph. We use the alternating direction method of multipliers to solve the semidefinite programming relaxations. Numerical results show that our bounds are currently the strongest bounds for the quadratic shortest path problem. We also present computational results on solving the quadratic shortest path problem using a branch and bound algorithm. Our algorithm computes a semidefinite programming bound in each node of the search tree, and solves instances with up to 1300 arcs in less than an hour (!).