Computing Treewidth via Exact and Heuristic Lists of Minimal Separators

Abstract

We develop practically efficient algorithms for computing the treewidth ${\mathrm{tw}}(G)$ of a graph G. The core of our approach is a new dynamic programming algorithm which, given a graph G, a positive integer k, and a set $\varDelta$ of minimal separators of G, decides if G has a tree-decomposition of width at most k of a certain canonical form that uses minimal separators only from $\varDelta$, in the sense that the intersection of every pair of adjacent bags belongs to $\varDelta$. This algorithm is used to show a lower bound of $k + 1$ on ${\mathrm{tw}}(G)$, setting $\varDelta$ to be the set of all minimal separators of cardinality at most k and to show an upper bound of k on ${\mathrm{tw}}(G)$, setting $\varDelta$ to be some, hopefully rich, set of such minimal separators. Combining this algorithm with new algorithms for exact and heuristic listing of minimal separators, we obtain exact algorithms for treewidth which overwhelmingly outperform previously implemented algorithms.