Inapproximability of shortest paths on perfect matching polytopes

Abstract

We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless ${\textsf {P}}={\textsf {NP}}$, there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on ${\textsf {P}}\ne {\textsf {NP}}$, this disproves a conjecture by Ito et al. (SIAM J Discrete Math 36(2):1102–1123, 2022). Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no polynomial-time algorithm computing a path of length at most $\left( \frac{1}{4}-o(1)\right) \log N / \log \log N$ between two vertices at distance two of the perfect matching polytope of an N-vertex bipartite graph. These results remain true if the bipartite graph is restricted to be of maximum degree three. The above has the following interesting implication for the performance of pivot rules for the simplex algorithm on simply-structured combinatorial polytopes: If ${\textsf {P}}\ne {\textsf {NP}}$, then for every simplex pivot rule executable in polynomial time and every constant $k \in {\mathbb {N}}$ there exists a linear program on a perfect matching polytope and a starting vertex of the polytope such that the optimal solution can be reached in two monotone non-degenerate steps from the starting vertex, yet the pivot rule will require at least k non-degenerate steps to reach the optimal solution. This result remains true in the more general setting of pivot rules for so-called circuit-augmentation algorithms.