On the maximum cardinality search lower bound for treewidth

Abstract

The Maximum Cardinality Search algorithm visits the vertices of a graph in an order such that at any point, a vertex is visited that has the largest number of visited neighbours. An MCS-ordering of a graph is an ordering of the vertices that can be generated by the Maximum Cardinality Search algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The MCSLB of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. Lucena [10] showed that the treewidth of a graph G is at least the MCSLB of any MCS-ordering of G. In this paper, we analyse the maximum MCSLB over all possible MCS-orderings of given graphs G. We give upper and lower bounds for this number for planar graphs. Given a graph G, it is NP-complete to determine if G has an MCS-ordering with MCSLB at least k, for any fixed k ≥ 7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete. We also propose and experimentally analysed some heuristics for the problem. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the maximum MCSLB that an MCS-ordering can obtain which appear to give results close to optimal on several graphs from real life applications. © Springer-Verlag 2004.