Hedgegraph Polymatroids

TL;DR

This paper introduces new measures of connectivity for hedgegraphs, a generalization of hypergraphs, and explores their structural and algorithmic properties through a polymatroidal framework, overcoming previous barriers in connectivity algorithms.

Contribution

It proposes two novel partition-based connectivity measures for hedgegraphs and studies their properties using polymatroids, enabling new algorithmic results.

Findings

New connectivity measures for hedgegraphs

Polymatroidal framework yields tractability results

Generalizes classical graph and hypergraph results

Abstract

Graphs and hypergraphs combine expressive modeling power with algorithmic efficiency for a wide range of applications. Hedgegraphs generalize hypergraphs further by grouping hyperedges under a color/hedge. This allows hedgegraphs to model dependencies between hyperedges and leads to several applications. However, it poses algorithmic challenges. In particular, the cut function is not submodular, which has been a barrier to algorithms for connectivity. In this work, we introduce two alternative partition-based measures of connectivity in hedgegraphs and study their structural and algorithmic aspects. Instead of the cut function, we investigate a polymatroid associated with hedgegraphs. The polymatroidal lens leads to new tractability results as well as insightful generalizations of classical results on graphs and hypergraphs.