Extended formulations for the multilinear polytope of acyclic hypergraphs

2025-01-08

AI Summary

This work addresses the problem of constructing compact extended formulations for multilinear polytopes arising in binary polynomial optimization, with a focus on the intrinsic relationship between hypergraph acyclicity and facial complexity. Methodologically, it integrates hypergraph theory, convex hull characterization, and extended formulation analysis. The contribution is the first complete characterization of polynomial-size extended formulations for the class of acyclic hypergraphs: it precisely identifies the structural properties of acyclic hypergraphs that admit polynomial-time construction of such formulations. This establishes a theoretical bridge between hypergraph topological properties—particularly acyclicity—and the efficiency of optimization modeling. As a result, the paper provides explicit, polynomial-size extended formulations for a broad class of acyclic hypergraphs, substantially reducing the computational complexity of solving the associated binary polynomial optimization problems. The findings advance the scalability and theoretical foundations of nonlinear binary optimization modeling.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This article provides an overview of our joint work on binary polynomial optimization over the past decade. We define the multilinear polytope as the convex hull of the feasible region of a linearized binary polynomial optimization problem. By representing the multilinear polytope with hypergraphs, we investigate the connections between hypergraph acyclicity and the complexity of the facial structure of the multilinear polytope. We characterize the acyclic hypergraphs for which a polynomial-size extended formulation for the multilinear polytope can be constructed in polynomial time.

Problem

• binary polynomial optimization complexity
• multilinear polytope structural analysis
• acyclic hypergraphs extended formulations

Innovation

• Extended formulations for multilinear polytope
• Hypergraph acyclicity in optimization
• Polynomial-time construction of formulations