最大边权k-plex划分问题的建模
Modeling the maximum edge-weight k-plex partitioning problem
摘要：
给定一个稀疏的无向图G，G的一个k重划分是它的一组节点的一个划分，使得每个组件是一个k重。一个节点子集S是一个k重的，如果在相关的诱导子图中每个节点的度至少是|S| -k。最大边权k-plex分割（Max-EkPP）问题是找到具有最大总权重的k-plex分割，其中分割的权重是解中边上的权重之和。当k=1时，划分中的所有组件都是团，并且问题变成众所周知的最大边权重团划分（Max-ECP）。然而，据我们所知，当k>1时，问题从未被建模过。实际上，关于k-丛的文献解决了在未加权图中搜索单个分量的问题。我们提出了一个多项式大小的整数线性规划制定的Max-EkPP问题，并考虑在模型中包含额外的拓扑约束。这些约束包括每个组件的容量下限和上极限以及最终解中组件数量的上界约束。所有这些特征都保持了模型的线性和初始多项式大小。我们还提出了计算测试，以显示模型的性能在不同的参数设置。这些测试诉诸于基准和真实世界的图表。
英文摘要：
Given a sparse undirected graph G with weights on the edges, a k-plex partition of G is a partition of its set of nodes such that each component is a k-plex. A subset of nodes S is a k-plex if the degree of every node in the associated induced subgraph is at least |S|-k. The maximum edge-weight k-plex partitioning (Max-EkPP) problem is to find a k-plex partition with maximum total weight, where the partition's weight is the sum of the weights on the edges in the solution. When k=1, all components in the partition are cliques and the problem becomes the well-known maximum edge-weight clique partitioning (Max-ECP). However, and to our best knowledge, when k>1, the problem has never been modeled. Actually, the literature on the k-plex addresses the search for a single component in an unweighted graph. We propose a polynomial size integer linear programming formulation for the Max-EkPP problem and consider the inclusion of additional topological constraints in the model. These constraints involve lower and upper limit capacity bounds in each component and upper bound constraints on the number of components in the final solution. All these characterizations preserve linearity and the initial polynomial size of the model. We also present computational tests in order to show the models' performance under different parameters' settings. These tests resort to benchmark and real-world graphs.
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