Dynamic Programming for Graphs on Surfaces

Abstract : We provide a framework for the design of $2^{\mathcal{O}(k)}\cdot n$ step dynamic programming algorithms for surface-embedded graphs on $n$ vertices of branchwidth at most $k$. Our technique applies to graph problems for which dynamic programming uses tables encoding set partitions. For general graphs, the best known algorithms for such problems run in $2^{\mathcal{O}(k\cdot \log k)}\cdot n$ steps. That way, we considerably extend the class of problems that can be solved by algorithms whose running times have a {\em single exponential dependence} on branchwidth, and improve the running time of several existing algorithms. Our approach is based on a new type of branch decomposition called {\em surface cut decomposition}, which generalizes sphere cut decompositions for planar graphs, and where dynamic programming should be applied for each particular problem. The construction of such a decomposition uses a new graph-topological tool called {\em polyhedral decomposition}. The main result is that if dynamic programming is applied on surface cut decompositions, then the time dependence on branchwidth is {\sl single exponential}. This fact is proved by a detailed analysis of how non-crossing partitions are arranged on surfaces with boundary and uses diverse techniques from topological graph theory and analytic combinatorics.
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Submitted on : Wednesday, December 30, 2009 - 5:21:40 PM
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Juanjo Rué, Ignasi Sau, Dimitrios M. Thilikos. Dynamic Programming for Graphs on Surfaces. [Research Report] RR-7166, INRIA. 2009, pp.39. ⟨inria-00443582⟩

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