Approximating Rank-Width and Clique-Width Quickly

Rank-width is defined by Seymour and the author to investigate clique-width; they show that graphs have bounded rank-width if and only if they have bounded clique-width. It is known that many hard graph problems have polynomial-time algorithms for graphs of bounded clique-width, however, requiring a given decomposition corresponding to clique-width (k-expression); they remove this requirement by constructing an algorithm that either outputs a rank-decomposition of width at most f(k) for some function f or confirms rank-width is larger than k in O(|V|9log |V|) time for an input graph G = (V,E) and a fixed k. This can be reformulated in terms of clique-width as an algorithm that either outputs a (21+f(k)–1)-expression or confirms clique-width is larger than k in O(|V|9log |V|) time for fixed k. In this paper, we develop two separate algorithms of this kind with faster running time. We construct a O(|V|4)-time algorithm with f(k) = 3k + 1 by constructing a subroutine for the previous algorithm; we may now avoid using general submodular function minimization algorithms used by Seymour and the author. Another one is a O(|V|3)-time algorithm with f(k) = 24k by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný.