Multiple Depot Ring Star Problem: A polyhedral study and exact algorithm

TL;DR

This paper introduces a new mixed integer linear programming formulation and a branch-and-cut algorithm for solving the Multiple Depot Ring-Star Problem, with extensive computational testing demonstrating its effectiveness.

Contribution

It provides the first polyhedral analysis and facet-inducing inequalities for the MDRSP, enhancing exact solution methods.

Findings

The proposed algorithm efficiently solves various test instances.

Polyhedral insights improve the solution process.

The method outperforms existing approaches on benchmark problems.

Abstract

The Multiple Depot Ring-Star Problem (MDRSP) is an important combinatorial optimization problem that arises in the context of optical fiber network design, and in applications pertaining to collecting data using stationary sensing devices and autonomous vehicles. Given the locations of a set of customers and a set of depots, the goal is to (i) find a set of simple cycles such that each cycle (ring) passes through a subset of customers and exactly one depot, (ii) assign each non-visited customer to a visited customer or a depot, and (iii) minimize the sum of the routing costs, i.e., the cost of the cycles and the assignment costs. We present a mixed integer linear programming formulation for the MDRSP and propose valid inequalities to strengthen the linear programming relaxation. Furthermore, we present a polyhedral analysis and derive facet-inducing results for the MDRSP. All these results are then used to develop a branch-and-cut algorithm to obtain optimal solutions to the MDRSP. The performance of the branch-and-cut algorithm is evaluated through extensive computational experiments on several classes of test instances.