Treedepth Bounds in Linear Colorings

Kun, Jeremy; O’Brien, Michael P.; Sullivan, Blair D.

doi:10.1007/978-3-030-00256-5_27

Cited by 4 publications

(6 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Attempts to get colorings with weaker structural properties but less colors were made e.g. in [46]. Unfortunately, recent experiments by O'Brien and Sullivan [66] indicate that the number of colors in a p-treedepth coloring in real-world graphs is too large for practical applications of this concept.…”

Section: Local Structuresmentioning

confidence: 99%

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

Nadara

Pilipczuk

Rabinovich

et al. 2019

ACM J. Exp. Algorithmics

View full text Add to dashboard Cite

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized coloring numbers and the property of being uniformly quasi-wide. We provide experimental evaluations of several algorithms that approximate these parameters on real-world graphs. On the theoretical side, we provide a new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and show a lower bound indicating that the guarantees of this algorithm are close to optimal in graph classes with fixed excluded minor.

show abstract

Section: Local Structuresmentioning

confidence: 99%

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

Nadara

Pilipczuk

Rabinovich

et al. 2019

ACM J. Exp. Algorithmics

View full text Add to dashboard Cite

show abstract

“…Clearly, each centered coloring is a linear coloring, but the minimum number of colors needed for a linear coloring can be much smaller than the treedepth of a graph. Kun et al [7] provided a polynomial relation between the treedepth and the minimum number of colors in a linear coloring; by replacing their usage of [6] by our result (and using an improved bound for the excluded grid theorem [1]) we obtain an improved bound.…”

Section: Theorem 13 ([6]mentioning

confidence: 88%

“…The result of Kawarabayashi and Rossman [6] has been also an important ingredient in the study of linear colorings [7]. A coloring α : V (G) → Z of a graph G is a linear coloring if for every (not necessarily induced) path P in G there exists a vertex v ∈ V (P ) of unique color α(v) on P .…”

Section: Theorem 13 ([6]mentioning

confidence: 99%

See 1 more Smart Citation

Improved bounds for the excluded-minor approximation of treedepth

Czerwiński¹,

Nadara²,

Pilipczuk³

2019

Preprint

View full text Add to dashboard Cite

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every positive integers a, b and a graph G, if the treedepth of G is at least Cab, then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least b as a subgraph.As a direct corollary, we obtain that every graph of treedepth Ω(k 3 ) is either of treewidth at least k, contains a subdivision of full binary tree of depth k, or contains a path of length 2 k . This improves the bound of Ω(k 5 log 2 k) of Kawarabayashi and Rossman [SODA 2018].We also show an application of our techniques for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t, one can in polynomial time compute a treedepth decomposition of G of width O(kt log 3/2 t). This improves upon a bound of O(kt 2 log t) stemming from a tradeoff between known results.The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d • log 3 ((1 + √ 5)/2).

show abstract

“…Some results in this paper appeared previously in WG 2018 [10]. Compared to the conference version, this version adds a polynomial treedepth upper bound for general graphs (Sect.…”

Section: Introductionmentioning

confidence: 93%

Polynomial Treedepth Bounds in Linear Colorings

et al. 2020

Self Cite

View full text Add to dashboard Cite

Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors have bounded treedepth. These colorings have an implicit tradeoff between the total number of colors used and the treedepth bound, and prior empirical work suggests that the former dominates the run time of existing algorithms in practice. We introduce p-linear colorings as an alternative to the commonly used p-centered colorings. They can be efficiently computed in bounded expansion classes and use at most as many colors as p-centered colorings. Although a set of k < p colors from a p-centered coloring induces a subgraph of treedepth at most k, the same number of colors from a p-linear coloring may induce subgraphs of larger treedepth. We establish a polynomial upper bound on the treedepth in general graphs, and give tighter bounds in trees and interval graphs via constructive coloring algorithms. We also give a co-NP-completeness reduction for recognizing p-linear colorings and discuss ways to overcome this limitation in practice.

show abstract

Treedepth Bounds in Linear Colorings

Cited by 4 publications

References 18 publications

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

Improved bounds for the excluded-minor approximation of treedepth

Polynomial Treedepth Bounds in Linear Colorings

Contact Info

Product

Resources

About