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

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.

• The paper establishes that computing the diameter of the bipartite perfect matching polytope is {if}p_2-complete, a complexity class even higher than NP-hard.
• Hardness is shown via a reduction from a universally quantified variant of Hamiltonian Cycle, encoded using intricate graph gadgets within the perfect matching structure.
• This work highlights the extreme computational difficulty of determining or approximating polytope diameters, even for structured cases, posing challenges for related algorithms like the simplex method.

The paper establishes that determining the diameter of the bipartite perfect matching polytope is not only computationally intractable—in fact, it is complete for the second level of the polynomial hierarchy—but in addition, the diameter (as well as the circuit diameter) is inapproximable within any factor strictly better than $1+\varepsilon$ (with a concrete, albeit very small, constant $\varepsilon$) unless $P=NP$.

The main technical contributions and ideas can be summarized as follows:

• Reduction from a $\forall\exists$-Hamiltonian Cycle Problem:

The core hardness proof is based on a reduction from a variant of the Hamiltonian cycle problem defined over planar directed graphs with bounded degree. In this variant, an instance consists of a directed graph together with a specified collection of arcs (formed by the choices from certain vertices having two outgoing edges), and the decision question asks whether for every pattern of choices among these arcs there exists a Hamiltonian cycle that respects the pattern. This problem is shown to be $\Pi^p_2$-hard.

• Gadget Constructions and Flip Sequences:

To translate an instance of the $\forall\exists$-Hamiltonian cycle problem into the language of perfect matchings on bipartite graphs (and hence into the setting of polytope diameters), the reduction constructs a bipartite graph $G_H$ that encodes the original decision problem in the structure of its perfect matchings. Several interlocking gadgets are introduced:

• Tower Gadgets:

  These are designed with a specified height (parameterized in the reduction) and serve to enforce lower bounds on the number of “flip” operations (alternating cycles) needed to alter the matching configuration locally. Technical lemmas show that any well-behaved flip sequence operating on a tower must contain at least a prescribed number of cycles (e.g., a tower of height $h$ needs at least $2h-2$ flips to change from one state to another) and that worst-case sequences of length exactly $2h$ are attainable.

• City Gadgets:

  Formed by the concatenation of many tower gadgets, these structures “force” any sufficiently short flip sequence between two perfect matchings to visit a large fraction of the city gadgets in a controlled (and regular) way. Regular cycles are defined as those that traverse every city gadget, and it is shown that only a very limited number of irregular cycles (which might “cheat” the desired structure) can appear in any short transition.

• Ladder and XOR Gadgets:

  The ladder gadgets exhibit slight asymmetric behavior when transformed via well-behaved cycles. By combining multiple ladder gadgets (within what is called a $\forall$-gadget) and linking them via XOR gadgets, the reduction forces an encoding of the “pattern” from the original $\forall\exists$-Hamiltonian cycle instance into the configuration of perfect matchings. In effect, the state (top-open versus bottom-open) in a ladder gadget corresponds to the selection of a particular arc among a pair.

• $\forall$-Gadgets:

  These gadgets aggregate many ladder gadgets and are connected to the remainder of $G_H$ in such a way that every regular cycle (i.e., cycle visiting all city gadgets) traverses each $\forall$-gadget in one of two well-defined ways. The chosen mode directly encodes the choice from the original instance. Detailed structural properties of these gadgets are established so that if the overall flip sequence (transforming one perfect matching to another) is short enough, then one can extract from it a Hamiltonian cycle in the original graph that respects the prescribed pattern.

  • Diameter and Inapproximability Results:

By showing that there is a flip sequence of length at most $4n^4+46n$ (where $n$ is the size parameter of the underlying instance) if and only if every pattern can be “completed” to form a Hamiltonian cycle in the original graph, the paper proves that the decision problem for the diameter of the bipartite perfect matching polytope is $\Pi^p_2$-complete. Furthermore, leveraging gap amplification arguments (via reductions that rely on the PCP theorem and bounded occurrence versions of 3SAT), the authors demonstrate that if one could approximate the diameter within a factor of $1+\varepsilon$ (for some constant $\varepsilon>0$), then one could distinguish satisfiable instances from those where no “almost-Hamiltonian” walk exists. In the reduction presented, the authors obtain a concrete value (on the order of $6.5\times 10^{-5}$, though they do not claim tightness) such that any better approximation would collapse $P$ and $NP$.

• Additional Insights and Technical Lemmas:

A careful analysis is provided that shows semi-default perfect matchings – matchings in which a predetermined subset of vertices is matched in a fixed way (corresponding to the “default” state of all gadgets) – are dense in the 1-skeleton of the matching polytope. This guarantees that every perfect matching is close (in terms of flip distance) to one with a canonical structure, simplifying the analysis of the global flip sequence. A series of lemmas rigorously bounds the number and structure of alternating cycles (or “flips”) that must occur in the gadget substructures when transforming one perfect matching to another.

Overall, the paper bridges techniques from combinatorial optimization, polyhedral theory, and computational complexity to give strong evidence that, even when restricted to a well-structured polytope (the perfect matching polytope associated with bipartite graphs), the problem of computing or even approximating its diameter is exceptionally hard. This result not only deepens the understanding of polytope diameters but also shows parallels with other notoriously hard combinatorial parameters (such as generalized Ramsey numbers).

The work concludes with several open questions, including whether the constant factor in the inapproximability result could be improved, whether the complexity can be extended to additional classes of polytopes (or even to the monotone version of the diameter), and how these hardness results might influence our approach to pivot-rule selection in the simplex method and related algorithmic paradigms in linear programming.

This contribution is significant for researchers interested in the polyhedral combinatorics of perfect matchings as well as those working on the complexity of geometric optimization problems.