Multi-constraint hypergraph partitioning is a generalization of balanced partitioning, where the vertex set of a hypergraph is partitioned such that the inter-block connectivity of hyperedges is minimized while balancing the vertices with regard to $d$ distinct constraints. A prominent class of applications is data distribution tasks, where this allows to achieve good load balance for $d$ different kinds of resources and simultaneously minimize the communication volume. Although the best approaches for single-constraint partitioning are usually complex (multilevel) algorithms with many components, we show that replacing only one component already leads to high-quality multi-constraint partitions: the rebalancing step, which restores balance for a partition that has (hopefully) small connectivity but violates the constraints. We design a multi-constraint rebalancing algorithm based on greedy local search, proving that balance is always restored for $d=2$ and bounded maximum weight. The key is to ensure monotonically decreasing global imbalance by choosing an imbalance metric where there is always a balance-improving move available. Integrating our algorithm into the state-of-the-art partitioner Mt-KaHyPar, we demonstrate an 11.5\,\% geometric mean connectivity reduction compared to the next best competitor (Metis) and better reliability regarding partition balance, even though the majority of inputs is outside of the theoretical guarantee.

翻译：多约束超图划分是平衡划分的推广，其目标是将超图顶点集划分为多个块，在满足$d$个不同约束的顶点平衡条件下，最小化超边间的块间连通性。一类典型应用是数据分配任务，通过该方法可在$d$类不同资源上实现良好负载均衡，同时最小化通信量。尽管单约束划分的最佳方法通常是包含多组件的复杂（多级）算法，但我们证明仅替换一个组件即可获得高质量的多约束划分：即再平衡步骤——该步骤用于恢复因连通性较小（有望如此）但违反约束的划分的平衡性。我们设计了一种基于贪心局部搜索的多约束再平衡算法，证明对于$d=2$且最大权重有界的情况，总能恢复平衡。关键在于通过选择一种始终存在平衡改进移动的失衡度量，确保全局失衡单调递减。将我们的算法集成到最先进的划分器Mt-KaHyPar中，与次优竞争者（Metis）相比，连通性几何平均降低了11.5%，且在划分平衡方面具有更高的可靠性——尽管大多数输入数据超出了理论保证范围。