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.

我们考虑在二分图的完美匹配多面体的骨架中寻找短路径的计算问题。我们证明，除非\({\textsf {P}}={\textsf {NP}}\)，否则不存在多项式时间算法可以计算距离为 2 的完美匹配多胞形的两个顶点之间的恒定长度路径二分图。以\({\textsf {P}}\ne {\textsf {NP}}\)为条件，这反驳了 Ito 等人的猜想。（SIAM J 离散数学 36(2):1102–1123, 2022）。假设指数时间假设，我们证明了更强的结果，即不存在计算最长路径的多项式时间算法\(\left( \frac{1}{4}-o(1)\right) \log N / \log \log N\)位于N顶点二分图的完美匹配多胞形的距离为 2 的两个顶点之间。如果二分图限制为最大三度，这些结果仍然成立。上述对于简单结构组合多胞形上的单纯形算法的主元规则的性能有以下有趣的含义：如果\({\textsf {P}}\ne {\textsf {NP}}\)，则对于每个单纯形在多项式时间内执行的枢轴规则和每个常数\(k \in {\mathbb {N}}\)都存在一个关于完美匹配多胞形和多胞形起始顶点的线性程序，使得可以在两个过程中达到最优解从起始顶点开始的单调非退化步骤，但枢轴规则将需要至少k 个非退化步骤才能达到最佳解决方案。这个结果在所谓的电路增强算法的枢轴规则的更一般设置中仍然成立。