Reduced cost-based ranking for generating promising subproblems

TL;DR

This paper introduces a reduced cost-based ranking method for generating promising subproblems in optimization, significantly improving the likelihood of finding optimal solutions quickly, especially in TSP variants.

Contribution

It presents a novel subproblem ranking technique using reduced costs within a Limited Discrepancy Search framework, enhancing solution efficiency.

Findings

Reduced costs provide highly accurate subproblem ranking.

Most optimal solutions are found in the first generated subproblem.

Method shows effectiveness on TSP and its variants.

Abstract

In this paper, we propose an effective search procedure that interleaves two steps: subproblem generation and subproblem solution. We mainly focus on the first part. It consists of a variable domain value ranking based on reduced costs. Exploiting the ranking, we generate, in a Limited Discrepancy Search tree, the most promising subproblems first. An interesting result is that reduced costs provide a very precise ranking that allows to almost always find the optimal solution in the first generated subproblem, even if its dimension is significantly smaller than that of the original problem. Concerning the proof of optimality, we exploit a way to increase the lower bound for subproblems at higher discrepancies. We show experimental results on the TSP and its time constrained variant to show the effectiveness of the proposed approach, but the technique could be generalized for other problems.