Dynamic Programming on Graphs with Bounded Treewidth

In this paper we study the complexity of graph decision problems, restricted to the class of graphs with treewidth k, (or equivalently, the class of partial k-trees), for fixed k. We introduce two classes of graph decision problems, LCC and ECC, and subclasses C-LCC, and C-ECC. We show that each problem in LCC (or C-LCC) is solvable in polynomial (O(n )) time, when restricted to graphs with fixed upper bounds on the treewidth and degree; and that each problem in ECC (or C-ECC) is solvable in polynomial (O(n )) time, when restricted to graphs with a fixed upper bound on the treewidth (with given corresponding tree-decomposition). Also, problems in C-LCC and C-ECC are solvable in polynomial time for graphs with a logarithmic treewidth, and given corresponding tree-decomposition, and in the case of C-LCC-problems, a fixed upper bound on the degree of the graph. Also, we show for a large number of graph decision problems, their membership in LCC, ECC, C-LCC and/or C-ECC, thus showing the existence of O(n ) or polynomial algorithms for these problems, restricted to the graphs with bounded tree width (and bounded degree). In several cases, C = 1, hence our method gives in these cases linear algorithms. For several NP-complete problems, and subclasses of the graphs with bounded treewidth, polynomial algorithms have been obtained. In a certain sense, the results in this paper unify these results.