Faster algorithms for vertex partitioning problems parameterized by clique-width

Abstract Many NP -hard problems, such as Dominating Set , are FPT parameterized by clique-width. For graphs of clique-width k given with a k-expression, Dominating Set can be solved in 4 k n O ( 1 ) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a ( 2 3 k − 1 ) -expression via rank-width and then solving Dominating Set using the ( 2 3 k − 1 ) -expression, the above algorithm will only give a runtime of 4 2 3 k n O ( 1 ) . There have been results which overcome this exponential jump; the best known algorithm can solve Dominating Set in time 2 O ( k 2 ) n O ( 1 ) by avoiding constructing a k-expression Bui-Xuan et al. (2013) [7] . We improve this to 2 O ( k log k ) n O ( 1 ) . Indeed, we show that for a graph of clique-width k, a large class of domination and partitioning problems (LC-VSP), including Dominating Set , can be solved in 2 O ( k log k ) n O ( 1 ) . Our main tool is a variant of rank-width using the rank of a 0–1 matrix over the rational field instead of the binary field.