## Approximating clique-width and branch-width (2006)

Venue: | JOURNAL OF COMBINATORIAL THEORY, SERIES B |

Citations: | 56 - 6 self |

### BibTeX

@ARTICLE{Oum06approximatingclique-width,

author = {Sang-il Oum and Paul Seymour},

title = {Approximating clique-width and branch-width},

journal = {JOURNAL OF COMBINATORIAL THEORY, SERIES B},

year = {2006},

volume = {96},

pages = {514--528}

}

### OpenURL

### Abstract

We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width”. Clique-width is a measure of the difficulty of decomposing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of the graph is called a “k-expression”. We find (for fixed k) an O(n 9 log n)-time algorithm that, with input an n-vertex graph, outputs either a (2 3k+2 − 1)expression for the graph, or a true statement that the graph has clique-width at least k + 1. (The best earlier algorithm algorithm, by Johansson [13], constructed a k log n-expression for graphs of clique-width at most k.) It was already known that several graph problems, NPhard on general graphs, are solvable in polynomial time if the input graph comes equipped with a k-expression (for fixed k). As a consequence of our algorithm, the same conclusion follows under the weaker hypothesis that the input graph has clique-width at most k (thus, we no longer need to be provided with an explicit k-expression). Another application is to the area of matroid branch-width. For fixed k, we find an O(n 4)time algorithm that, with input an n-element matroid in terms of its rank oracle, either outputs a branch-decomposition of width at most 3k − 1 or a true statement that the matroid has branch-width at least k + 1. The previous algorithm by Hliněn´y [11] was only for representable matroids.

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Citation Context ...omposition. • In particular, one of these problems is the problem of deciding whether a graph has tree-width at most k. Consequently, for fixed k there is a polynomial (indeed, linear) time algorithm =-=[1]-=- to test whether an input graph has tree-width at most k, and if so to output the corresponding decomposition. For inputs of bounded clique-width, less progress has so far been made. (We will define c... |

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Citation Context ... for purposes of comparison. Having bounded clique-width is more general than having bounded tree-width, in the following sense. Every graph G of tree-width at most k has clique-width at most O(2 k ) =-=[5, 7]-=-, and for such graphs (for k fixed) the clique-width of G can be determined in linear time [9]. No bound in the reverse direction holds, for there are graphs of arbitrary large tree-width with clique-... |

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Citation Context ...te f from f ∗ . For the first analysis, let γ be the time to compute f(X) for any set X. In this case we shall use f ∗ = fmin. To calculate fmin, we use the submodular function minimization algorithm =-=[12]-=-, whose running time is O(n 5 γ log M) where M is the maximum value of f and n = |V |. Thus, we can calculate fmin in O(n 5 γ log n) time. Finding a base X can be done by calculating f ∗ at most O(n) ... |

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Citation Context ...e-width. This class of problems is smaller than the corresponding set for tree-width, but still of interest. For instance, deciding whether the graph is Hamiltonian [22], finding the chromatic number =-=[14]-=-, and various partition problems [8] are solvable in polynomial time; and so is any problem that can be expressed in monadic second order logic with quantifications over vertices and vertex sets (MSO1... |

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Citation Context ...ection holds, for there are graphs of arbitrary large tree-width with clique-width at most k. (But, for fixed t, if G does not contain Kt,t as a subgraph, then the tree-width is at most 3k(t − 1) − 1 =-=[10]-=-.) The algorithmic situation with tree-width is as follows: • Numerous problems have been shown to be solvable in polynomial time when the input graph is presented together with a decomposition of bou... |

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Citation Context ...-width, in the following sense. Every graph G of tree-width at most k has clique-width at most O(2 k ) [5, 7], and for such graphs (for k fixed) the clique-width of G can be determined in linear time =-=[9]-=-. No bound in the reverse direction holds, for there are graphs of arbitrary large tree-width with clique-width at most k. (But, for fixed t, if G does not contain Kt,t as a subgraph, then the tree-wi... |

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Citation Context ...th a decomposition of bounded clique-width. This class of problems is smaller than the corresponding set for tree-width, but still of interest. For instance, deciding whether the graph is Hamiltonian =-=[22]-=-, finding the chromatic number [14], and various partition problems [8] are solvable in polynomial time; and so is any problem that can be expressed in monadic second order logic with quantifications ... |

2 |
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Citation Context ...input an n-vertex graph, outputs either a (2 3k+2 − 1)expression for the graph, or a true statement that the graph has clique-width at least k + 1. (The best earlier algorithm algorithm, by Johansson =-=[13]-=-, constructed a k log n-expression for graphs of clique-width at most k.) It was already known that several graph problems, NPhard on general graphs, are solvable in polynomial time if the input graph... |