Improved tree decomposition based algorithms for domination-like problems

Abstract

We present an improved dynamic programming strategy for DOMINATING SET and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4kn) algorithm for DOMINATING SET, where n is the number of nodes of the tree decomposition. This result improves the previously best known algorithm of Telle and Proskurowski running in time O(9kn). The key to our result is an argument on a certain “monotonicity” in the table updating process during dynamic programming. Moreover, various other domination-like problems as discussed by Telle and Proskurowski are treated with our technique.We gain improvements on the base of the exponential term in the running time ranging between 55% and 68% in most of these cases. These results mean significant breakthroughs concerning practical implementations.