Complexity of polytope diameters via perfect matchings

TL;DR

This paper proves that computing the circuit diameter of polytopes, including perfect matching polytopes, is strongly NP-hard, advancing understanding of the computational complexity of polytope diameter problems.

Contribution

It establishes the NP-hardness of computing the circuit diameter of polytopes and provides a graph-theoretic characterization of the monotone diameter of perfect matching polytopes.

Findings

Computing the circuit diameter of a polytope is strongly NP-hard.

Computing the combinatorial diameter of the perfect matching polytope is NP-hard.

The monotone diameter of perfect matching polytopes is also strongly NP-hard.

Abstract

The Circuit diameter of polytopes was introduced by Borgwardt, Finhold and Hemmecke as a fundamental tool for the study of circuit augmentation schemes for linear programming and for estimating combinatorial diameters. Determining the complexity of computing the circuit diameter of polytopes was posed as an open problem by Sanit\`a as well as by Kafer, and was recently reiterated by Borgwardt, Grewe, Kafer, Lee and Sanit\`a. In this paper, we solve this problem by showing that computing the circuit diameter of a polytope given in halfspace-description is strongly NP-hard. To prove this result, we show that computing the combinatorial diameter of the perfect matching polytope of a bipartite graph is NP-hard. This complements a result by Sanit\`a (FOCS 2018) on the NP-hardness of computing the diameter of fractional matching polytopes and implies the new result that computing the diameter of a $\{0,1\}$-polytope is strongly NP-hard, which may be of independent interest. In our second main result, we give a precise graph-theoretic description of the monotone diameter of perfect matching polytopes and use this description to prove that computing the monotone (circuit) diameter of a given input polytope is strongly NP-hard as well.