Abstract.In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(nk 2 3 k ) time and the problem to count the number of perfect matchings in O * (2 k ) time. Using a generalisation of fast subset convolution, we obtain faster algorithms for all [ρ, σ]-domination problems with finite or cofinite ρ and σ on tree decompositions. These include many well known graph problems. We give additional results on many more graph covering and partitioning problems.

We first propose a method, called "bottom-up method" that, informally, "propagates" improvement of the worst-case complexity for "sparse" instances to "denser" ones and we show an easy though non-trivial application of it to the MIN SET COVER problem. We then tackle MAX INDEPENDENT SET. Here, we propagate improvements of worst-case complexity from graphs of average degree d to graphs of average degree greater than d. Indeed, using algorithms for MAX INDEPENDENT SET in graphs of average degree 3, we successively solve MAX INDEPENDENT SET in graphs of average degree 4, 5 and 6. Then, we combine the bottom-up technique with measure and conquer techniques to get improved running times for graphs of maximum degree 5 and 6 but also for general graphs. The computation bounds obtained for MAX INDEPENDENT SET are O * (1.1571 n ), O * (1.1895 n ) and O * (1.2050 n ), for graphs of maximum (or more generally average) degree 4, 5 and 6 respectively, and O * (1.2114 n ) for general graphs. These results improve upon the best known results for these cases for polynomial space algorithms.

An edge dominating set in a graph G = (V, E) is a subset of the edges D ⊆ E such that every edge in E is adjacent or equal to some edge in D. The problem of finding an edge dominating set of minimum cardinality is NP-hard. We present a faster exact exponential time algorithm for this problem. Our algorithm uses O(1.3226 n ) time and polynomial space. The algorithm combines an enumeration approach of minimal vertex covers in the input graph with the branch and reduce paradigm. Its time bound is obtained using the measure and conquer technique. The algorithm is obtained by starting with a slower algorithm which is refined stepwise. In each of these refinement steps, the worst cases in the measure and conquer analysis of the current algorithm are reconsidered and a new branching strategy is proposed on one of these worst cases. In this way a series of algorithms appears, each one slightly faster than the previous, ending in the O(1.3226 n ) time algorithm. For each algorithm in the series, we also give a lower bound on its running time.We also show that the related problems: minimum weight edge dominating set, minimum maximal matching and minimum weight maximal matching can be solved in O(1.3226 n ) time and polynomial space using modifications of the algorithm for edge dominating set. In addition we consider the matrix dominating set problem which we solve in O(1.3226 n+m ) time and polynomial space, and the parametrised minimum weight maximal matching problem for which we obtain an O * (2.4178 k ) time algorithm.

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