Algorithms for Vertex Partitioning Problems on Partial k-Trees

• Authors: Jan Arne Telle;Andrzej Proskurowski
• Venue: SIAM Journal on Discrete Mathematics
• Year: 1997

In this paper, we consider a large class of vertex partitioning problems and apply to them the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications.We give a precise characterization of vertex partitioning problems, which include domination, coloring and packing problems, and their variants. Several new graph parameters are introduced as generalizations of classical parameters. This characterization provides a basis for a taxonomy of a large class of problems, facilitating their common algorithmic treatment and allowing their uniform complexity classification.We present a design methodology of practical solution algorithms for generally $\NP$-hard problems when restricted to partial k-trees (graphs with treewidth bounded by k). This "practicality" accounts for dependency on the parameter k of the computational complexity of the resulting algorithms.By adapting the algorithm design methodology on partial k-trees to vertex partitioning problems, we obtain the first algorithms for these problems with reasonable time complexity as a function of treewidth. As an application of the methodology, we give the first polynomial-time algorithm on partial k-trees for computation of the Grundy number.