Approximating rank-width and clique-width quickly

Sang-Il Oum
2008 ACM Transactions on Algorithms
Rank-width was defined by Oum and Seymour [2006. Approximating clique-width and branchwidth. J. Combin. Theory Ser. B 96, 4, 514-528] 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
more » ... ct an O(|V | 4 )-time algorithm with f (k) = 3k + 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) = 24k by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005. A parametrized algorithm for matroid branch-width. SIAM J. Comput. 35, 2, 259-277]. Finally we construct an O(|V | 3 )-time algorithm with f (k) = 3k − 1 by combining the ideas of above two cited papers. Corollary 1.1. Let k be a fixed positive integer. There is an O(|V (G)| 3 )-time algorithm that either outputs an (8 k −1)-expression of an input graph G or confirms that the clique-width of G is larger than k. Courcelle and Oum  showed that, for each k, there is a formula of modulo-2 counting monadic second-order logic (C 2 MS logic) expressing that the rank-width is at most k. C 2 MS logic is an extension of first order logic, which allows set variables denoting sets of objects (in our case, sets of vertices) and the set predicate