The higher-order correlation clustering problem for a graph $G$ and costs associated with cliques of $G$ consists in finding a clustering of $G$ so as to minimize the sum of the costs of those cliques whose nodes all belong to the same cluster. To tackle this NP-hard problem in practice, local search heuristics have been proposed and studied in the context of applications. Here, we establish partial optimality conditions for cubic correlation clustering, i.e., for the special case of at most 3-cliques. We define and implement algorithms for deciding these conditions and examine their effectiveness numerically, on two data sets.

翻译：对于图$G$及其关联团成本的更高阶相关聚类问题，其目标在于寻找$G$的一种聚类划分，使得所有节点均属于同一聚类的团成本之和最小。为在实践中处理这一NP难问题，已有研究在应用背景下提出并分析了局部搜索启发式算法。本文针对立方相关聚类——即最多涉及3-团的特例——建立了部分最优性条件。我们定义并实现了判定这些条件的算法，并在两个数据集上数值检验了其有效性。