Ear-decompositions and the complexity of the matching polytope

TL;DR

This paper investigates the complexity of the matching polytope via ear-decompositions, providing efficient algorithms for certain cases and extending results to binary matroids, thereby advancing understanding of h-perfection and related properties.

Contribution

It introduces a polynomial-time method to decide if the matching polytope has a simple description and extends the analysis to binary matroids, showing fixed-parameter tractability.

Findings

Deciding whether β(G) ≤ 1 can be done efficiently using ear-decompositions.

The approach simplifies testing h-perfection in line-graphs.

Computing β is fixed-parameter tractable for binary matroids.

Abstract

The complexity of the matching polytope of graphs may be measured with the maximum length $\beta$ of a starting sequence of odd ears in an ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its facets are defined by 2-connected factor-critical graphs, which have an odd ear-decomposition (according to a theorem of Lov\'asz). In particular, $\beta(G) \leq 1$ if and only if the matching polytope of the graph $G$ is completely described by non-negativity, star and odd-circuit inequalities. This is essentially equivalent to the h-perfection of the line-graph of $G$, as observed by Cao and Nemhauser. The complexity of computing $\beta$ is apparently not known. We show that deciding whether $\beta(G)\leq 1$ can be executed efficiently by looking at any ear-decomposition starting with an odd circuit and performing basic modulo-2 computations. Such a greedy-approach is surprising in view of the complexity of the problem in more special cases by Bruhn and Schaudt, and it is simpler than using the Parity Minor Algorithm. Our results imply a simple polynomial-time algorithm testing h-perfection in line-graphs (deciding h-perfection is open in general). We also generalize our approach to binary matroids and show that computing $\beta$ is a Fixed-Parameter-Tractable problem (FPT).