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Approximating clique-width and branch-width. (English) Zbl 1114.05022
The authors show that, for fixed $$k$$, there is an algorithm that with input an $$n$$-vertex graph $$G$$, either decides that $$G$$ has clique-width at least $$k+1$$ or outputs a decomposition of $$G$$ with clique-width at most $$2^{3k+2}-1$$. The running time of the algorithm is shown to be $$\mathbb{O}(n^9 \log{n})$$. The main tool for this algorithm is branch-width, which was introduced by N. Robertson and P. D. Seymour [J. Comb. Theory, Ser. B 52, 153-190 (1991; Zbl 0764.05069)]. The present authors develop a general algorithm to approximate the branch-width of certain symmetric submodular functions. Then the “rank-width” of a graph is defined and it is shown that bounded clique-width implies bounded rank-width and vice versa. Consequently clique-width can be approximated in polynomial time. Additionally the algorithm is applied to matroids: For fixed $$k$$ there is an algorithm which, with input an $$n$$-element matroid $$M$$ in terms of its rank oracle, either decides that $$M$$ has branch-width at least $$k+1$$, or outputs a branch-decomposition for $$M$$ of width at most $$3k-1$$. The running time of the algorithm is $$\mathbb{O}(n^{3.5})$$.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
algorithm; rank-width; submodular function; matroid
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