Submission Track B : exact algorithm for Minimum Fill - In ( aka Chordal Completion )

Our implementation follows the exact algorithm of Fomin et al. [3] which is based on listing the minimal separators, the potential maximal cliques, and then performing dynamic programming based on those structures [1, 2]. The algorithm by Fomin et al. refines the approach of Bouchitté and Todinca on some minor aspects (only considering inclusion-minimal minimal separators, trying only the so-called full blocks associated to a minimal separator) and also lists the potential maximal cliques in a different way. We realized that this theoretically better enumeration of potential maximal cliques was in practice slower, so for this part we follow Bouchitté and Todinca’s algorithm. This algorithm can be seen as FPT parameterized by the number of potential maximal cliques. Its dependency in the parameter is polynomial (even linear) which does not imply that the problem is polytime solvable since the number of potential maximal cliques (and minimal separators) can be as big as exponential in the number of vertices of the graph. It is noteworthy that this nice and deep theory (of potential maximal cliques) initiated by Bouchitté and Todinca seems to perform, as is, better than more straightforward approaches (such as the naive branching on all chordalizations of a long induced cycle). This is not always the case and more often than not the intricate and theoretically best algorithms perform quite poorly in practice. This sets Bouchitté and Todinca’s algorithm is the regarded class of theoretical work with a direct practical value.