Factorized binary polynomial optimization

TL;DR

This paper introduces a new formulation for binary polynomial optimization called factorized binary polynomial optimization, and provides a strongly polynomial time algorithm for fixed r, expanding tractability and improving on existing methods.

Contribution

The paper proposes a novel factorized formulation for binary polynomial optimization and develops an algorithm that solves it in strongly polynomial time for fixed r, broadening the class of tractable problems.

Findings

Algorithm solves factorized binary polynomial optimization in strongly polynomial time for fixed r.

Improves on the state-of-the-art for quadratic objective functions.

Applicable to binary tensor factorization and Boolean tensor problems with fixed rank.

Abstract

In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial optimization. In this formulation, we assume that the variables are partitioned into a fixed number of sets, and that the objective function is written as a sum of r products of linear functions, each one involving only variables in one set of the partition. Our main result is an algorithm that solves factorized binary polynomial optimization in strongly polynomial time, when r is fixed. This result provides a vast new class of tractable instances of binary polynomial optimization, and it even improves on the state-of-the-art for quadratic objective functions, both in terms of generality and running time. We demonstrate the applicability of our result through the binary tensor factorization problem, which arises in mining discrete patterns in data, and that contains as a special case the rank-1 Boolean tensor factorization problem. Our main result implies that these problems can be solved in strongly polynomial time, if the input tensor has fixed rank, and a rank factorization is given. For the rank-1 Boolean matrix factorization problem, we only require that the input matrix has fixed rank.