Catalan structures and dynamic programming in H-minor-free graphs

Abstract

We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex H-minor-free graph G contains a path of length k in 2 O ( √ k) · n O (1) steps. Our approach builds on a combination of Demaine-Hajiaghayi's bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of Hminor-free graphs. This result is used to bound the number of ways vertex disjoint paths can be routed through the separators of such decompositions. The proof is based on several structural theorems from the Graph Minors series of Robertson and Seymour. With a slight modification, similar combinatorial and algorithmic results can be derived for many other problems. Our approach can be viewed as a general framework for obtaining time 2 O ( √ k) · n O (1) algorithms on H-minor-free graph classes.

Graph Minors and Parameterized Algorithm Design

“…There is a quite extended bibliography on how to do fast dynamic programming on graphs of bounded treewidth; as a sample of this, we just mention [5,6,9,11,13,16,22,35,35,36,37,37,38,114,120,120]. t: Bounds are much better for the function t. For most natural graph parameters, it holds that t(k) = O(k) while for some of them, including tw and pw, it holds that t(k) = Θ(k).…”

Planar k-Path in Subexponential Time and Polynomial Space

“…If n ≤ 2 √ k apply the algorithm from Lemma 2 to solve the problem in time 2 O( √ k log 2 k) n O(1) and using polynomial space. If, on the other hand n ≥ 2 √ k the standard dynamic programming algorithm on graphs of bounded treewidth that uses at most 2 O( √ k) n O(1) time and space [7], runs in polynomial time and space. This concludes the proof.…”

Deterministic Single Exponential Time Algorithms for Connectivity Problems Parameterized by Treewidth

“…Early results on the special case of Hminor-free graphs were given in [9]. A positive answer to the question was found by Cygan et al [8] using a randomized approach termed "Cut & Count": It provided a transformation of the natural certificates to "cut-certificates" transforming the connectivity requirement into a locally checkable requirement.…”

“…E.g., we unify algorithms for Feedback Vertex Set [6] and k-Path [2] and generalize algorithms for restricted inputs such as H-minor-free (e.g. [9]) or bounded degree graphs (e.g. [10]).…”