Beating the SDP bound for the floor layout problem: A simple combinatorial idea

TL;DR

This paper introduces a simple combinatorial method to obtain strong dual bounds for floor layout problems, outperforming SDP relaxations and being computationally easier for challenging instances.

Contribution

The paper presents a novel combinatorial bounding technique that surpasses SDP bounds for floor layout problems, offering a practical alternative.

Findings

Bounds are at least as good as SDP relaxations.

The method is significantly stronger for difficult instances.

Computationally easier to implement and solve.

Abstract

For many mixed-integer programming (MIP) problems, high-quality dual bounds can be obtained either through advanced formulation techniques coupled with a state-of-the-art MIP solver, or through semidefinite programming (SDP) relaxation hierarchies. In this paper, we introduce an alternative bounding approach that exploits the "combinatorial implosion" effect by solving portions of the original problem and aggregating this information to obtain a global dual bound. We apply this technique to the one-dimensional and two-dimensional floor layout problems and compare it with the bounds generated by both state-of-the-art MIP solvers and by SDP relaxations. Specifically, we prove that the bounds obtained through the proposed technique are at least as good as those obtained through SDP relaxations, and present computational results that these bounds can be significantly stronger and easier to compute than these alternative strategies, particularly for very difficult problem instances.