Hypergraph Cuts with Edge-Dependent Vertex Weights

TL;DR

This paper introduces a novel framework for hypergraph cuts that incorporates edge-dependent vertex weights, enabling more accurate modeling of vertex importance within hyperedges and improving cut solutions.

Contribution

The authors develop EDVWs-based splitting functions, prove their submodularity, and show how to reduce hypergraphs to graphs for efficient minimum s-t cut computation.

Findings

EDVWs-based splitting functions outperform traditional methods.

The reduction to graphs preserves cut properties.

Sparsification accelerates the algorithms on reduced graphs.

Abstract

We develop a framework for incorporating edge-dependent vertex weights (EDVWs) into the hypergraph minimum s-t cut problem. These weights are able to reflect different importance of vertices within a hyperedge, thus leading to better characterized cut properties. More precisely, we introduce a new class of hyperedge splitting functions that we call EDVWs-based, where the penalty of splitting a hyperedge depends only on the sum of EDVWs associated with the vertices on each side of the split. Moreover, we provide a way to construct submodular EDVWs-based splitting functions and prove that a hypergraph equipped with such splitting functions can be reduced to a graph sharing the same cut properties. In this case, the hypergraph minimum s-t cut problem can be solved using well-developed solutions to the graph minimum s-t cut problem. In addition, we show that an existing sparsification technique can be easily extended to our case and makes the reduced graph smaller and sparser, thus further accelerating the algorithms applied to the reduced graph. Numerical experiments using real-world data demonstrate the effectiveness of our proposed EDVWs-based splitting functions in comparison with the all-or-nothing splitting function and cardinality-based splitting functions commonly adopted in existing work.