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

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 semi-definite 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.

KEYWORDS:

• Layout
• integer programming

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Combinatorial implosion is a more optimistic corollary to the well-known combinatorial explosion effect, which is used throughout the folklore to describe the difficulty of many discrete optimization problems.

Additional information

Funding

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship [grant number 1122374], [grant number CMMI-1351619].