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Inapproximability of shortest paths on perfect matching polytopes

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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.

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Notes

  1. We note that the sole purpose of splitting vertices into binary trees is to restrict the maximum degree of the graph, the remainder of the proof is only based on the 4-cycles in the middle of the gadgets.

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All authors contributed to the study. All authors read and approved the final manuscript. This work was initiated during a stay of the first author as a guest professor at ETH in May-July 2022. He wishes to thank Prof. Emo Welzl for his hospitality during this stay. The second author was supported by an ETH Postdoctoral Fellowship. The authors also wish to thank the referees for their careful reading and comments.

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Correspondence to Jean Cardinal.

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Cardinal, J., Steiner, R. Inapproximability of shortest paths on perfect matching polytopes. Math. Program. (2023). https://doi.org/10.1007/s10107-023-02025-4

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