SAT-Encodings for Treecut Width and Treedepth

The decomposition of graphs is a prominent algorithmic task with numerous applications in computer science. A graph decomposition method is typically associated with a width parameter (such as treewidth) that indicates how well the given graph can be decomposed. Many hard (even #P-hard) algorithmic problems can be solved efficiently if a decomposition of small width is provided; the runtime, however, typically depends exponentially on the decomposition width. Finding an optimal decomposition is itself an NP-hard task. In this paper we propose, implement, and test the first practical decomposition algorithms for the width parameters tree-cut width and treedepth. These two parameters have recently gained a lot of attention in the theoretical research community as they offer the algorithmic advantage over treewidth by supporting so-called fixed-parameter algorithms for certain problems that are not fixed-parameter tractable with respect to treewidth. However, the existing research has mostly been theoretical. A main obstacle for any practical or experimental use of these two width parameters is the lack of any practical or implemented algorithm for actually computing the associated decompositions. We address this obstacle by providing the first practical decomposition algorithms.

Our approach for computing treecut width and treedepth decompositions is based on efficient encodings of these decomposition methods to the propositional satisfiability problem (SAT). Once an encoding is generated, any satisfiability solver can be used to find the decomposition. This allows us to leverage the surprising power of todays state-of-the art SAT solvers. The success of SAT-based decomposition methods crucially depends on the used characterisation of the decomposition method, as not every characterisation is suitable for that task. For instance, the successful leading SAT encoding for treewidth is based on a characterisation of treewidth in terms of elimination orderings. For treecut width and treedepth, however, we propose new characterisations that are based on sequences of partitions of the vertex set, a method that was pioneered for clique-width. We implemented and systematically tested our encodings on various benchmark instances, including famous named graphs and random graphs of various density. It turned out that for the considered width parameters, our partition-based SAT encoding even outperforms the best existing SAT encoding for treewidth.

We hope that our encodings—which we will make publicly available—will stimulate the experimental research on the algorithmic use of treecut width and tree depth, and thus will help to bride the gap between theoretical and experimental research. For future work we propose to scale our approach to larger graphs by means of SAT-based local improvement, a method that have been recently shown successful for the width parameters treewidth and branchwidth.