SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning

TL;DR

This paper develops SDP relaxations and an augmented Lagrangian cutting plane algorithm for the Quadratic Cycle Cover Problem, integrating reinforcement learning for improved bounds and solution strategies.

Contribution

It introduces novel SDP relaxations, an efficient facial reduction technique, and a cutting plane algorithm combined with reinforcement learning for the QCCP.

Findings

Our SDP bounds outperform existing methods.

The proposed algorithm efficiently solves large SDP relaxations.

Reinforcement learning techniques improve upper bounds.

Abstract

We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting plane framework by utilizing Dykstra's projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting planes. Computational results show that our SDP bounds and our efficient cutting plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment.