Optimization Bounds from the Branching Dual

TL;DR

This paper introduces a novel branching duality method to derive strong bounds for discrete optimization problems, improving existing bounds without relying on integer programming models, demonstrated on the minimum bandwidth problem.

Contribution

It develops a duality-based approach for bounding discrete problems, utilizing partial branching trees and a local search heuristic to enhance bounds.

Findings

Substantially improves known bounds for the minimum bandwidth problem.

Outperforms depth-first and breadth-first branching in bound quality.

Heuristic effectively explores neighboring partial trees for better bounds.

Abstract

We present a general method for obtaining strong bounds for discrete optimization problems that is based on a concept of branching duality. It can be applied when no useful integer programming model is available, and we illustrate this with the minimum bandwidth problem. The method strengthens a known bound for a given problem by formulating a dual problem whose feasible solutions are partial branching trees. It solves the dual problem with a "worst-bound" local search heuristic that explores neighboring partial trees. After proving some optimality properties of the heuristic, we show that it substantially improves known combinatorial bounds for the minimum bandwidth problem with a modest amount of computation. It also obtains significantly tighter bounds than depth-first and breadth-first branching, demonstrating that the dual perspective can lead to better branching strategies when the object is to find valid bounds.