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

TL;DR

This paper proves that computing the diameter of bipartite perfect matching polytopes is $\Pi^p_2$-hard, answering a 30-year-old question, and establishes inapproximability results for the circuit diameter of these polytopes.

Contribution

It establishes the $\Pi^p_2$-completeness of computing the bipartite perfect matching polytope diameter and proves inapproximability bounds, advancing understanding of polytope diameter complexity.

Findings

Computing the bipartite perfect matching polytope diameter is $\Pi^p_2$-hard.

The problem is in $\Pi^p_2$, confirming $\Pi^p_2$-completeness.

The circuit diameter cannot be approximated within a factor better than $(1 + \varepsilon)$.

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.