Finding Optimal Triangulations Parameterized by Edge Clique Cover

Abstract

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.