Approximating rank-width and clique-width quickly

Rank-width was defined by Oum and Seymour [2006] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f ( k ) for some function f or confirms that rank-width is larger than k in time O (| V | 9 log | V |) for an input graph G = ( V , E ) and a fixed k . We develop three separate algorithms of this kind with faster running time. We construct an O (| V | 4 )-time algorithm with f ( k ) = 3 k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O (| V | 3 )-time algorithm with f ( k ) = 24 k , achieved by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005]. Finally we construct an O (| V | 3 )-time algorithm with f ( k ) = 3 k − 1 by combining the ideas of the two previously cited papers.