Separable and Equatable Hypergraphs

TL;DR

This paper explores the properties of separable and equatable hypergraphs, establishing their exclusive nature, and investigates the complexity of recognizing separability in different classes of hypergraphs and matroids.

Contribution

It introduces the concepts of separable and equatable hypergraphs, proves their exclusive relationship, and analyzes the complexity of recognizing separability in various hypergraph classes.

Findings

Every hypergraph is either separable or equatable, but not both.

Recognition of separability is exponential time for paving matroids with independence oracles.

Recognition is polynomial time for binary matroids with independence oracles.

Abstract

We consider the class of {\em separable} $k$-hypergraphs, which can be viewed as uniform analogs of threshold Boolean functions, and the class of {\em equatable} $k$-hypergraphs. We show that every $k$-hypergraph is either separable or equatable but not both. We raise several questions asking which classes of equatable (and separable) hypergraphs enjoy certain appealing characterizing properties, which can be viewed as uniform analogs of the $2$-summable and $2$-monotone Boolean function properties. In particular, we introduce the property of {\em exchangeability}, and show that all these questioned characterizations hold for graphs, multipartite $k$-hypergraphs for all $k$, paving $k$-matroids and binary $k$-matroids for all $k$, and $3$-matroids, which are all equatable if and only if they are exchangeable. We also discuss the complexity of deciding if a hypergraph is separable, and in particular, show that it requires exponential time for paving matroids presented by independence oracles, and can be done in polynomial time for binary matroids presented by such oracles.