Citation Context

...l implementations. Cook and Seymour [10, 11] used branch decompositions to solve the ring routing problem, related to the design of reliable cost-effective SONET networks and to solving TSP (see also =-=[7, 19]-=-). In theory, there is not a big difference between tree-width and branch-width based algorithms. However in practice, branch-width is sometimes easier to use. The question due to Bodlaender (private ...

Citation Context

...ing efficient algorithms for graph problems on classes of graphs of bounded tree-width. (See e.g. [5] and references therein.) Similarly, efficient algorithms can sometimes be found for planar graphs =-=[13, 2]-=- or more general classes of graphs, for instance classes of graphs of bounded local tree-width [14], or graph classes excluding a minor (see e.g. [10] and references therein). Another interesting exam...

Citation Context

...ree decompositions or on the related branch decompositions was introduced by Dorn [9], who employed fast matrix multiplication. If the input graph is planar, then other improvements are possible, see =-=[10,12]-=-. In [10], Dorn showed amongst others, that Dominating Set for planar graphs with a given tree decomposition of width k can be solved in O∗(3k) time. Some of these results can be generalised to graphs...

Citation Context

...g our approach can be specialized for planar graphs such that one might get rid of the log k-factor in the exponent of Corollary 5.1. The details, however, are tricky and we only refer to Dorn et al. =-=[13]-=- for more on that. 6 Conclusion Our main focus in this work was on problem kernels of linear size. As Fellows [16] puts in his recent keynote speech, kernelization is a “humble strategy for coping wit...

Citation Context

... on graphs of bounded genus. However, for some problems, like Hamiltonian Cycle, "-embedded Graph TSP, Max Leaf Tree, and Steiner Tree, branchwidth arguments do not provide us with time 2O( ! n) ·nO(1) algorithms. The reason is that all these problems are “non-local” and despite many attempts, no time 2o(!(G) log !) · nO(1) algorithm solving these problems on graphs of branchwidth at most ! is known. Recently, it was observed by several authors that if a graph G is not only of branchwidth at most ! but is also planar, then for a number of “non-local” problems the log ! overhead can be removed [4, 5], resulting in time 2O( ! n) · nO(1) algorithms on planar graphs. Similar result can be obtained by making use of separators [2]. It is a common belief that almost every technique working on planar graphs can be extended on graphs embedded on a surface of bounded genus. However, this is not always a straightforward task. The main di!culty in generalizing planar graph techniques [2, 4, 5] to graphs of bounded genus is that all these techniques are based on partitioning a graph embedded on a plane by a closed curve into smaller pieces. Deineko et al. use cyclic separators of triangulations [2], ...

Citation Context

...d this bound to O(4 ℓ ) · n O(1) by using several tricks, but also were the first to give a subexponential fixed parameter algorithm for Planar Dominating Set. Recently there have been several papers =-=[11, 4, 8, 12, 13]-=-, showing that for planar graphs or graphs of bounded genus the base of the exponent in the running time of these algorithms could be improved by instead doing dynamic programming along a branch decom...

Citation Context

...bgraphs 2and count subgraphs of G isomorphic to H in time 2 O(k) n and enumerate ω subgraphs in time 2 O(k) n + O(ωk). It is worth mentioning that for k-Longest Path on planar graphs, the authors of =-=[12]-=- give the first algorithm with time complexity subexponential in the parameter value. The algorithm has running time 2 O(√ k) n + O(n 3 ), employing the techniques Bidimensionality and topology-exploi...