Partitioning the Bags of a Tree Decomposition Into Cliques

TL;DR

This paper introduces the concept of clique-partitioned treewidth, explores the computational complexity of partitioning bags into cliques, and presents heuristics and an exact algorithm to approximate solutions, demonstrating its potential for real-world networks.

Contribution

It defines clique-partitioned treewidth, proves the NP-hardness of computing clique partitions, and develops algorithms to approximate solutions, advancing the understanding of this new parameter.

Findings

Branch-and-bound algorithm is efficient as a baseline.

Heuristics produce near-optimal solutions.

Clique-partitioned treewidth is promising for high-clustering graphs.

Abstract

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.