The Fine Details of Fast Dynamic Programming over Tree Decompositions

We study implementation details for dynamic programming over tree decompositions. Firstly, a fact that is overlooked in many papers and books on this subject is that it is not clear how to test adjacency between two vertices in time bounded by a function of k, where k is the width of the given tree decomposition. This is necessary to obtain linear time dynamic programming algorithms. We address this by giving a simple O(kn) time and space preprocessing procedure that enables adjacency testing in time O(k), where n is the number of vertices of the graph. Secondly, we show that a large class of NP-hard problems can be solved in time O(qn), where q is the natural size of the dynamic programming tables. The key improvement is that we avoid a polynomial factor in k. This holds for all problems that can be formulated as a Min Weight Homomorphism problem: given a (large) graph G on n vertices and a (small) graph H on q vertices, with integer vertex and edge weights, is there a homomorphism from G to H with total (vertex and edge image) weight at most M? This result implies e.g. O(2n) algorithms for Max Independent Set and Max Cut, and a O(qn) algorithm for qColorability. The table building techniques we develop are also useful for many other problems.