Dynamic Programming for Graphs on Surfaces

Juanjo Rué 1 Ignasi Sau 2 Dimitrios M. Thilikos 2, 3
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where stan-dard dynamic programming runs in 2 O(k·log k) · n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called surface cut decom-position, generalizing sphere cut decompositions of planar graphs which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2 O(k) · n steps. That way, we considerably extend the class of prob-lems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.
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Juanjo Rué, Ignasi Sau, Dimitrios M. Thilikos. Dynamic Programming for Graphs on Surfaces. ACM Transactions on Algorithms, Association for Computing Machinery, 2014, 10 (2), pp.Article No. 8. ⟨10.1145/2556952⟩. ⟨lirmm-01083690⟩

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