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 20(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 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 20(k).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. © 2010 Springer-Verlag Berlin Heidelberg.