Projection-width as a structural parameter for discrete separable optimization

TL;DR

This paper introduces the concept of projection-width, a structural parameter for systems of separable constraints, enabling polynomial-time solutions for a broad class of discrete nonlinear optimization problems.

Contribution

It defines projection-width and demonstrates that problems with bounded projection-width can be solved efficiently, unifying various tractability results across multiple optimization domains.

Findings

Polynomial-time algorithms for problems with bounded projection-width

Generalization of tractability results in integer linear and polynomial optimization

Unification of diverse problem classes under a single structural framework

Abstract

While several classes of integer linear optimization problems are known to be solvable in polynomial time, far fewer tractability results exist for integer nonlinear optimization. In this work, we narrow this gap by identifying a broad class of discrete nonlinear optimization problems that admit polynomial-time algorithms. Central to our approach is the notion of projection-width, a structural parameter for systems of separable constraints, defined via branch decompositions of variables and constraints. We show that several fundamental discrete optimization and counting problems can be solved in polynomial time when the projection-width is polynomially bounded, including optimization, counting, top-k, and weighted constraint violation problems. Our results subsume and generalize some of the strongest known tractability results across multiple research areas: integer linear optimization, binary polynomial optimization, and Boolean satisfiability. Although these results originated independently within different communities and for seemingly distinct problem classes, our framework unifies and significantly generalizes them under a single structural perspective.