An Experimental Approach to Exact and Random Boolean-Widths and Their Comparison with Other Width Parameters

Abstract

Parameterized complexity is an exemplary approach that extracts and exploits the power of the hidden structures of input instances to solve hard problems. The tree-width (⁠|$tw$|⁠), path-width (⁠|$pathw$|⁠), branch-width (⁠|$bw$|⁠), clique-width (⁠|$cw$|⁠), rank-width (⁠|$rw$|⁠) and boolean-width (⁠|$boolw$|⁠) are some width measures of graphs that are used as parameters. Applications of these width parameters show that dynamic programming algorithms based on a path, tree or branch decomposition can be an alternative to other existing techniques for solving hard combinatorial problems on graphs. A large number of the linear- or polynomial-time fixed parameter tractability algorithms for problems on graphs start by computing a decomposition tree of the graph with a small width. The focus of this paper is to study the exact and random boolean-widths for special graphs, real-world graphs and random graphs, as well as to check their competency compared with several other existing width parameters. In our experiments, we use graphs from TreewidthLIB, which is a set of named graphs and random graphs generated by the Erdös–Rényi model. Until now, only very limited experimental work has been carried out to determine the exact and random boolean-widths of graphs. Moreover, there are no approximation algorithms for computing the near-optimal boolean-width of a given graph. The results of this paper demonstrate that the boolean-width can be used not only in theory but also in practice and is competitive with other width parameters for real graphs.