Dynamic Programming for Graphs on Surfaces

TL;DR

This paper introduces a new framework for dynamic programming on surface-embedded graphs, achieving single-exponential time complexity in branchwidth by using topological and combinatorial tools.

Contribution

It develops a novel surface cut decomposition technique that generalizes sphere cut decompositions, enabling more efficient algorithms for surface-embedded graphs.

Findings

Dynamic programming runs in 2^{O(k)} n steps on surface-embedded graphs.

Introduces surface cut decomposition with favorable combinatorial properties.

Unifies and improves previous algorithms for problems on surface-embedded graphs.

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^{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 decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, 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 problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.