Projection-width: a unifying structural parameter for separable discrete optimization

2025-11-04

AI Summary

This paper addresses discrete optimization and counting problems over separable constraint systems. We introduce *projection width* as a unifying structural parameter, defined via a variable-constraint branch decomposition tree. Leveraging dynamic programming and the separability assumption, we are the first to incorporate projection width into this setting—enabling exact polynomial-time algorithms for optimization, counting, Top-$k$ enumeration, and weighted constraint violation minimization whenever the projection width is polynomially bounded. Our framework unifies and generalizes strong tractability results from diverse domains, including integer linear programming and propositional satisfiability (SAT). Crucially, it reveals a broad, natural class of nonlinear discrete problems that admit efficient exact solutions, thereby substantially expanding the known landscape of polynomial-time solvable problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce the notion of projection-width 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. These include optimization, counting, top-k, and weighted constraint violation. Our results identify a broad class of tractable nonlinear discrete optimization and counting problems. Even when restricted to the linear setting, they subsume and substantially extend 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.

Problem

• Introduces projection-width for separable constraint systems
• Solves discrete optimization and counting with polynomial projection-width
• Unifies tractability results across integer and nonlinear optimization

Innovation

• Projection-width defined via branch decompositions
• Polynomial-time solving for bounded projection-width
• Unifies nonlinear and linear discrete optimization problems