Dynamic Programming for Graphs on Surfaces

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 standard dynamic programming runs in $2^{\mathcal{O}(k\cdot \log k)}\cdot n$ 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 decomposition, capturing how partial solutions can be arranged on a surface. Then we use singularity analysis over expressions obtained by the symbolic method to prove 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 $2^{\mathcal{O}(k)}\cdot n$ steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve all previous results in this direction.

Research supported by the European Research Council under the EC’s 7th Framework Programme, ERC grant agreement 208471 - ExploreMaps project, the Israel Science Foundation, grant No. 1249/08, and the project “Kapodistrias” (AΠ 02839/28.07.2008) of the National and Kapodistrian University of Athens.