Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems

Given a graph G we provide dynamic programming algorithms for many locally checkable vertex subset and vertex partitioning problems. Their runtime is polynomial in the number of equivalence classes of problem-specific equivalence relations on subsets of vertices, defined on a given decomposition tree of G. Using these algorithms all these problems become solvable in polynomial time for many well-known graph classes like interval graphs and permutation graphs (Belmonte and Vatshelle (2013)  [1]). Given a decomposition of boolean-width k we show that the algorithms will have runtime O(n42O(k2)), providing the first large class of problems solvable in fixed-parameter single-exponential time in boolean-width.