Beyond hypergraph acyclicity: limits of tractability for pseudo-Boolean optimization

2024-10-30

arXiv.org

AI Summary

This work investigates the computational complexity of minimizing pseudo-Boolean optimization (PBO) problems over signed hypergraphs, aiming to delineate the tractability boundary. Method: We establish, for the first time, best-case hardness for PBO on bounded-rank hypergraphs; introduce the “nest-set gap” — a novel hypergraph measure quantifying deviation from acyclicity — and prove that bounded nest-set gap implies polynomial-size extended formulations; and rigorously derive superpolynomial lower bounds on treewidth and exponential extended formulation complexity via a synthesis of hypergraph theory, treewidth analysis, pseudo-Boolean polyhedral modeling, and extended formulation complexity theory. Contribution/Results: We uncover deep structural connections between PBO complexity and hypergraph parameters—including intersection graph treewidth and nest-set gap; and design the first polynomial-time algorithm for a broad class of hypergraphs containing β-cycles, substantially expanding the known tractable regime for PBO.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper, we study the problem of minimizing a polynomial function with literals over all binary points, often referred to as pseudo-Boolean optimization. We investigate the fundamental limits of computation for this problem by providing new necessary conditions and sufficient conditions for tractability. On the one hand, we obtain the first intractability results, in the best-case sense, for pseudo-Boolean optimization problems on signed hypergraphs with bounded rank, in terms of the treewidth of the intersection graph. Namely, first, under some mild assumptions, we show that for every sequence of hypergraphs indexed by the treewidth and with bounded rank, the complexity of solving the associated pseudo-Boolean optimization problem grows super-polynomially in the treewidth. Second, we show that any hypergraph of bounded rank is the underlying hypergraph of some signed hypergraph for which the corresponding pseudo-Boolean polytope has an exponential extension complexity in the treewidth. On the other hand, we introduce the nest-set gap, a new hypergraph-theoretic notion that enables us to define a notion of"distance"from the hypergaph acyclicity. We prove that if this distance is bounded, the pseudo-Boolean polytope admits a polynomial-size extended formulation. This in turn enables us to obtain a polynomial-time algorithm for a large class of pseudo-Boolean optimization problems whose underlying hypergraphs contain beta-cycles.

Problem

• Pseudo-Boolean Optimization
• Computational Complexity
• Hypergraph Optimization

Innovation

• Nest Gap
• Pseudo-Boolean Optimization
• Signed Graphs