Solving Unconstrained Binary Polynomial Programs with Limited Reach

25 Pages Posted: 3 Aug 2023

See all articles by Jens Vinther Clausen

Jens Vinther Clausen

Technical University of Denmark

Yves Crama

University of Liège

Richard Martin Lusby

Technical University of Denmark

Elisabeth Rodríguez-Heck

RWTH Aachen University

Stefan Ropke

Technical University of Denmark

Abstract

Unconstrained Binary Polynomial Programs (UBPs) are a class of optimization problems relevant in a broad array of fields. In this paper, we examine an example from communication engineering, namely low autocorrelation binary sequences and propose a new dynamic programming approach that is particularly effective on UBP instances that have a limited so-called reach, which is the maximum difference between any two variable indices in the problem. Based on the reach, the dynamic programming approach decomposes the problem into a number of overlapping stages that can be solved in parallel. On a set of publicly available low autocorrelation binary sequence problems, we demonstrate the superiority of the approach by showing that the method solves to optimality for the first time several previously unsolved instances. In particular, we provide a direct comparison between the proposed method and a modern version of a previously proposed dynamic program for UBPs. We give a detailed analysis of the connection between the two different algorithms and demonstrate that the advantage of the proposed dynamic program is in its ability to implicitly identify the multilinear polynomials that are required in the recursive steps of the two dynamic programs. For perspective, a comparison to several other methods is also provided.

Keywords: unconstrained binary polynomial program, Dynamic programming, low autocorrelated binary sequences

Suggested Citation

Clausen, Jens Vinther and Crama, Yves and Lusby, Richard Martin and Rodríguez-Heck, Elisabeth and Ropke, Stefan, Solving Unconstrained Binary Polynomial Programs with Limited Reach. Available at SSRN: https://ssrn.com/abstract=4530363 or http://dx.doi.org/10.2139/ssrn.4530363

Jens Vinther Clausen

Technical University of Denmark ( email )

Anker Engelunds Vej 1
Building 101A
Lyngby, 2800
Denmark

Yves Crama

University of Liège ( email )

Richard Martin Lusby

Technical University of Denmark ( email )

Anker Engelunds Vej 1
Building 101A
Lyngby, 2800
Denmark

Elisabeth Rodríguez-Heck

RWTH Aachen University ( email )

Templergraben 55
52056 Aachen, 52056
Germany

Stefan Ropke (Contact Author)

Technical University of Denmark ( email )

Anker Engelunds Vej 1
Building 101A
Lyngby, 2800
Denmark

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