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 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 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.

Hitting Minors on Bounded Treewidth Graphs. I. General Upper Bounds

“…Proof. The proof is an adaptation of the proof of Theorem 3.5 to more general surfaces, using some of the tools developed in [44]. A noose of a Σ g -embedded graph G is any G-normal O-arc of Σ.…”

“…According to [44], given a Σ g -embedded graph G, a polyhedral decomposition of G can be constructed in O(n 3 ) steps. We now consider such a polyhedral decomposition of G, we root it at the union of the apex set A and the intersection of its bag with its parent bag in T and, as we did the beginning of the proof of Theorem 3.5, we process the bags of (X , T ) in a bottom-up fashion.…”

“…This reduces the proof to the case where the input graph, we call it G, contains a set A of O(g) apex vertices such that are G A = G \ A is a polyhedrally Σ g -embedded graph for some g ≤ g. We also see G A as a Σ g -embedded t-boundaried graph G A whose boundary has size at most 2. To deal with this, we will use an analogue of sphere-cut decompositions for graphs that are polyhedrally embedded in higher-genus surfaces, called surface-cut decomposition, introduced in [44]. This requires some definitions in order to extend the concept of ∆-embedded t-boundaried graph to surfaces.…”

“…The concept of a surface-cut decomposition was introduced in [44] in order to accelerate dynamic programming algorithms in surfaces. We next present this concept, adapted to the terminology that we introduced above.…”

“…We prove that, if we use sphere-cut decompositions and we apply essentially the same dynamic programming algorithm discussed above, the number of representatives can be upper-bounded by the number of (unlabeled) planar graphs on O(tw) vertices, which are 2 O(tw) many [48]. With some more technical details, we extend this single-exponential algorithm to graphs embedded in surfaces by using surface-cut decompositions, introduced by Rué et al [44].…”