Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game

TL;DR

This paper introduces a unified approach for hypergraph partitioning using polymatroidal cut functions, achieving improved approximation ratios and faster algorithms through metric relaxations and an enhanced cut-matching game framework.

Contribution

It proposes a new class of cut functions and extends the cut-matching game to hypergraphs, enabling near-linear time algorithms with strong approximation guarantees.

Findings

Achieves an $O(\sqrt{ ext{log} n})$-approximation for hypergraph partitioning.

Develops an almost-linear time $O( ext{log} n)$-approximation algorithm.

Introduces a generalized cut-matching game with relaxed actions for hypergraph partitioning.

Abstract

Despite there being significant work on developing spectral, and metric embedding based approximation algorithms for hypergraph generalizations of conductance, little is known regarding the approximability of hypergraph partitioning objectives beyond this. This work proposes algorithms for a general model of hypergraph partitioning that unifies both undirected and directed versions of many well-studied partitioning objectives. The first contribution of this paper introduces polymatroidal cut functions, a large class of cut functions amenable to approximation algorithms via metric embeddings and routing multicommodity flows. We demonstrate an $O(\sqrt{\log n})$-approximation, where $n$ is the number of vertices in the hypergraph, for these problems by rounding relaxations to metrics of negative-type. The second contribution of this paper generalizes the cut-matching game framework of Khandekar et. al. to tackle polymatroidal cut functions. This yields the first almost-linear time $O(\log n)$-approximation algorithm for standard versions of undirected and directed hypergraph partitioning. A technical consequence of our construction is that a cut-matching game which greatly relaxes the set of allowed actions for both players can be used to partition hypergraphs with negligible impact on the approximation ratio. We believe this to be of independent interest.