The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs

TL;DR

This paper extends the concept of tight cut decomposition from graphs to hypergraphs, showing that for uniform hypergraphs, the decomposition is unique and can be used to analyze their perfect matching polytopes efficiently.

Contribution

It generalizes tight cut decomposition to hypergraphs, proves uniqueness for uniform hypergraphs, and demonstrates polynomial-time recognition of tight cuts in these hypergraphs.

Findings

Tight cut decomposition is not unique for general hypergraphs.

Uniqueness of the decomposition holds for a slight generalization of uniform hypergraphs.

Recognition of tight cuts in uniformable hypergraphs is polynomial time solvable.

Abstract

The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lov\'asz and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lov\'asz on the tight cut decomposition is that the list of bricks obtained is the same independent of the choice of tight cuts made during the tight cut decomposition procedure. This implies that finding a tight cut decomposition is polynomial time equivalent to finding a single tight cut. We generalise the notions of a tight cut, a tight cut contraction and a tight cut decomposition to hypergraphs. By providing an example, we show that the outcome of the tight cut decomposition on general hypergraphs is no longer unique. However, we are able to prove that the uniqueness of the tight cut decomposition is preserved on a slight generalisation of uniform hypergraphs. Moreover, we show how the tight cut decomposition leads to a decomposition of the perfect matching polytope of uniformable hypergraphs and that the recognition problem for tight cuts in uniformable hypergraphs is polynomial time solvable.