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.

翻译：尽管已有大量工作为电导率的超图推广开发了基于谱方法和度量嵌入的近似算法，但对于超越这一范畴的超图划分目标的可近似性仍知之甚少。本文提出了一种通用超图划分模型算法，该模型统一了众多经典有向与无向划分目标。本文的首个贡献是引入多拟阵割函数——这类割函数可通过度量嵌入与多商品流路由实现近似算法。通过将松弛舍入到负型度量，我们针对此类问题证明了$O(\sqrt{\log n})$近似比（其中$n$为超图顶点数）。第二个贡献将Khandekar等人的割匹配游戏框架推广至多拟阵割函数，首次为无向与有向超图划分的标准版本提供了近线性时间$O(\log n)$近似算法。我们的构造带来的技术成果是：一种极大放宽双方玩家允许动作集的割匹配游戏，可在对近似比影响可忽略的前提下实现超图划分。我们认为这一发现具有独立研究价值。