The Fine Details of Fast Dynamic Programming over Tree Decompositions

Abstract

AbstractWe 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(q k + 1 n), where q k + 1 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 (di)graph G on n vertices and a (di)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(2k n) algorithms for Max Independent Set and Max Cut, and a O(q k + 1 n) algorithm for q-Colorability. The table building techniques we develop are also useful for many other problems.KeywordsDynamic ProgrammingLeaf NodeDynamic Programming AlgorithmTree DecompositionLinear Time AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.