Computing the Polytope Diameter is Even Harder than NP-hard (Already for Perfect Matchings)
Lasse Wulf
TL;DR
The paper proves that computing the diameter of the bipartite perfect matching polytope is $Π^p_2$-complete, answering a 30-year-old question and showing that diameter computation lies strictly beyond NP unless the polynomial hierarchy collapses. It extends the hardness to the circuit diameter and proves a strong inapproximability bound, indicating no $(1+ε)$-approximation is possible in poly-time for some small $ε>0$. The main method reduces a $ orall∃$-HamCycle problem to BPM-Diameter-Decision via a sophisticated gadget construction (towers, cities, ladders, XORs, and ∀-gadgets) built into a graph $G_H$, tying short flip sequences to pattern-respecting Hamiltonian cycles. The results imply significant gaps between NP-hardness and higher-level PH hardness for polytope diameters and challenge prospects for practical diameter-based optimization via standard encodings, while also deriving corollaries for circuit-diameter computations. The work leaves open questions about constant-factor approximations, planarity-restricted instances, and extensions to other polytopes.
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. $Π^p_2$-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. Sanità [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 Nöbel [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 $Π^p_2$-hard. Since the corresponding decision problem is also trivially contained in $Π^p_2$, 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 Nöbel and Steiner. It is the first known inapproximability result for the circuit diameter, and extends Sanità's inapproximability result of the diameter to the totally unimodular case.