Polynomial Treedepth Bounds in Linear Colorings
Abstract
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 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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Sep 03, 2020
- Source ID
- 10.1007/s00453-020-00760-0
Entities
People
- Blair D Sullivan
- Jeremy Kun
- Marcin Pilipczuk
- Michael P. O’brien
Organizations
- Defense Advanced Research Projects Agency
- Gordon and Betty Moore Foundation
- University of Warsaw