A sufficiently fast algorithm for finding close to optimal clique trees
Artificial Intelligence
Efficient Approximation for Triangulation of Minimum Treewidth
UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
Safe Reduction Rules for Weighted Treewidth
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
A New Lower Bound for Tree-Width Using Maximum Cardinality Search
SIAM Journal on Discrete Mathematics
Dynamic programming, tree-width and computation on graphical models
Dynamic programming, tree-width and computation on graphical models
A spectral lower bound for the treewidth of a graph and its consequences
Information Processing Letters
A complete anytime algorithm for treewidth
UAI '04 Proceedings of the 20th conference on Uncertainty in artificial intelligence
Journal of Combinatorial Theory Series B
Safe Reduction Rules for Weighted Treewidth
Algorithmica
New lower and upper bounds for graph treewidth
WEA'03 Proceedings of the 2nd international conference on Experimental and efficient algorithms
Treewidth lower bounds with brambles
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Degree-Based treewidth lower bounds
WEA'05 Proceedings of the 4th international conference on Experimental and Efficient Algorithms
New upper bound heuristics for treewidth
WEA'05 Proceedings of the 4th international conference on Experimental and Efficient Algorithms
Treewidth computations II. Lower bounds
Information and Computation
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The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS 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 visited degree of an MCS-ordering @j of G is the maximum visited degree over all vertices v in @j. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree. We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k@?N. Given a graph G, it is NP-complete to determine if its maximum visited degree is 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. In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.