A Faster Parameterized Algorithm for Treedepth

Abstract

The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which-given as input an n-vertex graph, a tree decomposition of the graph of width w, and an integer t-decides Treedepth, i.e. whether the treedepth of the graph is at most t, in time 2 O(wt) · n. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time 2 2 O(t) · n and 2 O(t 2 ) · n, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nešetřil as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed t) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of 2 O(t log t) · n for the same algorithm. * Research funded by DFG-Project RO 927/13-1 "Pragmatic Parameterized Algorithms".

Losing Treewidth by Separating Subsets

“…The problem of finding a tree (or path) decomposition of width k parameterized by the width t of the input tree decomposition has been studied by [BK91] who gave a 2 O(tk+t log t)) n-time algorithm for the pathwidth case and a 2 O(t 2 k) n-time algorithm for the treewidth case. A 2 O(tk) n-time algorithm for finding a treedepth decomposition of width k given a tree decomposition of width t has been obtained by [RRVS14]. We slightly modify these procedures to handle vertex deletion and use them over the precomputed tree decomposition of width O(k + p).…”

“…For Treewidth / Pathwidth / Treedepth Vertex Deletion, we inject the bounds from Lemma 13 into Corollary 12. To handle Path Transversal observe that the k-path-free graphs have treedepth bounded by k and therefore also treewidth bounded by k [RRVS14]. There is an exact algorithm for k-Path Transversal with running time f k (n, p) = O(k p n), where p is the bound on the solution size [Lee18].…”

Uniform Kernelization Complexity of Hitting Forbidden Minors

“…Like in prior work on Treewidth-η Deletion [11], adding such edges does not change the answer to the problem. We then apply an algorithm by Reidl et al [34] to compute an approximate treedepth-η modulator S of the resulting graph. The remainder of the algorithm strongly exploits the structure of the boundedtreedepth graph G − S. By combining separators for vertices that are not linked through many disjoint paths, we compute a small set Y such that all the bounded-treedepth connected components of G − (S ∪ Y ) have a special structure: their neighborhood in S forms a clique, while they have less than η neighbors in Y .…”

“…Reidl et al [34] gave an algorithm with running time 2 O(t 2 ) · n to test whether the treedepth of graph is at most t. Gajarský et al [22] obtained meta-theorems for kernelization when parameterized by a treedepth-η modulator. They showed, for example, that problems satisfying certain technical conditions admit linear kernels on hereditary graphs of bounded expansion when parameterized by the size of a treedepth-η modulator.…”

“…The nontrivial algorithm tasks that have to be done for one iteration of the algorithm are determining the treedepth of the subgraph induced by a subtree of F , and finding the maximum number of internally vertex-disjoint paths in some subgraph. For every fixed η, the first can be done in polynomial (even linear) time, using for example the FPT algorithm of Reidl et al [34]. It is well known that the number of internally vertex-disjoint paths can be computed in polynomial time using flow techniques; see, for example, Schrijver [38,Chapter 9].…”