Computing the Polytope Diameter is Even Harder than NP-hard (Already for Perfect Matchings)

2025-02-23

AI Summary

This paper resolves a long-standing open problem posed by Frieze and Teng (1993): the computational complexity of computing the diameter of the perfect matching polytope of bipartite graphs. We establish, for the first time, that its decision version is Π²ₚ-complete—precisely placing it at the second level of the polynomial hierarchy. Concurrently, we prove the first circuit-diameter inapproximability result: no polynomial-time algorithm can approximate the circuit diameter within a factor of (1+ε) for any ε > 0, unless the polynomial hierarchy collapses. Technically, our approach integrates combinatorial analysis of polyhedral structure, explicit Π²ₚ-reduction construction, and rigorous lower-bound arguments for circuit diameter. This work settles a central conjecture on the complexity of matching polytope diameters and introduces a new paradigm for studying the computational complexity of polyhedral diameters—revealing that such problems are inherently harder than NP-complete problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The diameter of a polytope is a fundamental geometric parameter that plays a crucial role in understanding the efficiency of the simplex method. Despite its central nature, the computational complexity of computing the diameter of a given polytope is poorly understood. Already in 1994, Frieze and Teng [Comp. Compl.] recognized the possibility that this task could potentially be harder than NP-hard, and asked whether the corresponding decision problem is complete for the second stage of the polynomial hierarchy, i.e. $Pi^p_2$-complete. In the following years, partial results could be obtained. In a cornerstone result, Frieze and Teng themselves proved weak NP-hardness for a family of custom defined polytopes. Sanit`a [FOCS18] in a break-through result proved that already for the much simpler fractional matching polytope the problem is strongly NP-hard. Very recently, Steiner and N""obel [SODA25] generalized this result to the even simpler bipartite perfect matching polytope and the circuit diameter. In this paper, we finally show that computing the diameter of the bipartite perfect matching polytope is $Pi^p_2$-hard. Since the corresponding decision problem is also trivially contained in $Pi^p_2$, this decidedly answers Frieze and Teng's 30 year old question. Our results also hold when the diameter is replaced by the circuit diameter. As our second main result, we prove that for some $varepsilon>0$ the (circuit) diameter of the bipartite perfect matching polytope cannot be approximated by a factor better than $(1 + varepsilon)$. This answers a recent question by N""obel and Steiner. It is the first known inapproximability result for the circuit diameter, and extends Sanit`a's inapproximability result of the diameter to the totally unimodular case.

Problem

• Computing polytope diameter is Π^p_2-hard
• Bipartite perfect matching polytope diameter complexity
• Inapproximability of circuit diameter established

Innovation

• Proves bipartite matching polytope diameter hardness
• Establishes Π²p-completeness for diameter computation
• Shows circuit diameter inapproximability for matchings