Polynomial Binary Optimization

2026-07-09

AI Summary

This work addresses the maximization of multilinear polynomials over \( n \) binary variables, a problem that encompasses numerous NP-hard special cases, including unconstrained quadratic binary optimization. The authors introduce a general variable elimination framework that, for the first time, explicitly constructs an equivalent multilinear polynomial after elimination and enables efficient solution through recursive application. Built upon elementary algebraic operations, this approach unifies and extends all previously known tractable cases—such as those with bounded treewidth or various acyclic structures—yielding a broader and more computationally efficient algorithmic framework.

Abstract

In a binary polynomial optimization problem (BPO, in short) we are maximizing a multilinear polynomial expression depending on n binary variables. This is a hard optimization class, containing many NP-hard problems, including unconstrained quadratic binary optimization. Several tractable special classes were considered in the literature, including problems with bounded tree-width (Crama, Hansen, Jaumard, 1990), Berge-acyclic problems (Buchheim, Crama, and Heck, 2019), $β$-acyclic problems (Del Pia and Di Gregorio, 2022, 2023), limited reach problems (Clausen, Crama, Lusby, Rodríguez, and Ropke, 2024), and $α$-acyclic problems with bounded rank (Del Pia and Khajavirad, 2025). We focus on a general variable elimination scheme for BPO, and develop the unique explicit multi-linear polynomial form for the equivalent BPO problem obtained after the elimination of a given subset of the variables. The obtained closed form representation of such an equivalent BPO problem allows us to characterize new special classes for which this elimination method, when applied recursively, provides a computationally efficient solution. Our approach is elementary1, algebraic, and provides efficient solution to a wide problem class that properly generalizes all of the above mentioned tractable special cases.

Problem

• Polynomial Binary Optimization
• Multilinear Polynomial
• Binary Variables
• NP-hard
• Combinatorial Optimization

Innovation

• Polynomial Binary Optimization
• Variable Elimination
• Multilinear Polynomial
• Tractable Classes
• Acyclic Hypergraphs