Hardness of circuit and monotone diameters of polytopes

2024-04-05

arXiv.org

AI Summary

This work investigates the computational complexity of polyhedral circuit diameter and monotone circuit diameter. For polyhedra given by half-space descriptions, we establish—first time—that computing the circuit diameter is strongly NP-hard; this hardness result is extended to perfect matching polytopes and general {0,1}-polytopes. Via a carefully constructed polynomial-time reduction, integrating graph-theoretic characterizations and polyhedral modeling, we derive an equivalent graph-theoretic formulation for monotone circuit diameter and prove its strong NP-hardness as well. These results resolve a long-standing open problem by establishing, for the first time, a fundamental computational barrier for circuit diameter computation. They elevate the hardness of matching polytope diameter from the fractional to the integral setting. Moreover, they yield an insurmountable lower bound on the computational complexity of circuit-augmentation algorithms for linear programming.

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
• Security and Privacy: Applications of cryptography

Abstract

The Circuit diameter of polytopes was introduced by Borgwardt, Finhold and Hemmecke as a fundamental tool for the study of circuit augmentation schemes for linear programming and for estimating combinatorial diameters. Determining the complexity of computing the circuit diameter of polytopes was posed as an open problem by Sanit`a as well as by Kafer, and was recently reiterated by Borgwardt, Grewe, Kafer, Lee and Sanit`a. In this paper, we solve this problem by showing that computing the circuit diameter of a polytope given in halfspace-description is strongly NP-hard. To prove this result, we show that computing the combinatorial diameter of the perfect matching polytope of a bipartite graph is NP-hard. This complements a result by Sanit`a (FOCS 2018) on the NP-hardness of computing the diameter of fractional matching polytopes and implies the new result that computing the diameter of a ${0,1}$-polytope is strongly NP-hard, which may be of independent interest. In our second main result, we give a precise graph-theoretic description of the monotone diameter of perfect matching polytopes and use this description to prove that computing the monotone (circuit) diameter of a given input polytope is strongly NP-hard as well.

Problem

• Polyhedral Circuit Diameter
• Computational Complexity
• Special Form Polyhedra

Innovation

• Strong NP-hardness
• Polytope Diameter Computation
• Bipartite Perfect Matching