Practical algorithms on partial k-trees with an application to domination-like problems

Many NP-hard problems on graphs have polynomial, in fact usually linear, dynamic programming algorithms when restricted to partial k-trees (graphs of treewidth bounded by k), for fixed values of k. We investigate the practicality of such algorithms, both in terms of their complexity and their derivation, and account for the dependency on the treewidth k. We define a general procedure to derive the details of table updates in the dynamic programming solution algorithms. This procedure is based on a binary parse tree of the input graph. We give a formal description of vertex subset optimization problems in a class that includes several variants of domination, independence, efficiency and packing. We give algorithms for any problem in this class, which take a graph G, integer k and a width k tree-decomposition of G as input, and solve the problem on G in O(n24k) steps.