The diameter of the fractional matching polytope and its hardness implications

TL;DR

This paper proves that calculating the diameter of the fractional matching polytope is strongly NP-hard and APX-hard, even for simple structures, by providing an exact characterization of its diameter.

Contribution

It establishes the computational hardness of diameter calculation for the fractional matching polytope and offers an exact characterization of its diameter.

Findings

Diameter computation is strongly NP-hard for the fractional matching polytope.

Finding the maximum shortest path on the polytope's 1-skeleton is APX-hard.

An exact characterization of the polytope's diameter is provided.

Abstract

The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard. In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the \emph{fractional matching} polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an \emph{exact characterization} of the diameter of the fractional matching polytope, that is of independent interest.