We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions. Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.

翻译：我们考虑树宽度的一种变体，称为团划分树宽度，其中每个袋子被划分为团。这一研究受到近期基于类似参数针对各类问题开发固定参数可解算法的启发。本文首次尝试计算团划分树分解。我们的重点在于计算团划分的子问题，即对于给定树分解中的每个袋子，计算其诱导子图的一个最优的团划分。目标是尽可能最小化团大小（加1）的乘积。我们证明该问题是NP难的。同时，我们描述了四种启发式方法以及一种精确的分支定界算法。评估结果表明，分支定界求解器的效率足以作为良好的基准。此外，我们的启发式方法能够获得接近最优的解。作为额外成果，我们的算法能够计算现实世界网络的团划分树宽度的首个上界。与传统树宽度的比较表明，对于具有高聚类特征的图，团划分树宽度是一个有前景的参数。