# A Note on Exact Algorithms for Vertex Ordering Problems on Graphs

@article{Bodlaender2011ANO, title={A Note on Exact Algorithms for Vertex Ordering Problems on Graphs}, author={Hans L. Bodlaender and F. Fomin and Arie M. C. A. Koster and Dieter Kratsch and Dimitrios M. Thilikos}, journal={Theory of Computing Systems}, year={2011}, volume={50}, pages={420-432} }

In this note, we give a proof that several vertex ordering problems can be solved in O∗(2n) time and O∗(2n) space, or in O∗(4n) time and polynomial space. The algorithms generalize algorithms for the Travelling Salesman Problem by Held and Karp (J. Soc. Ind. Appl. Math. 10:196–210, 1962) and Gurevich and Shelah (SIAM J. Comput. 16:486–502, 1987). We survey a number of vertex ordering problems to which the results apply.

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This thesis focuses on three different vertex ordering problems, considers a variant of Directed Feedback Arc Set problem with applications in computational biology, and presents an efficient heuristic algorithm for this problem and a general algorithmic strategy for the problem.

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This paper uses a new set of constraints for modeling linear ordering problems on graphs using Integer Linear Programming to propose new ILP formulations for well-known linear ordering optimization problems, namely the Pathwidth, Cutwidth, Bandwidth, SumCut and Optimal Linear Arrangement problems.

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Experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential-time algorithms using exponential space or using only polynomial space are given and it is shown that with a more complicated algorithm using balanced separators, Treewidth can be computed in O*(2.9512n) time and polynometric space.

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This is the first algorithm with running time better than the straightforward O*(2n) time for computing the directed pathwidth of a digraph with n vertices, and as a special case, it computes the path width of an undirected graph in the same time.

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