TY - GEN

T1 - Efficient fully dynamic elimination forests with applications to detecting long paths and cycles

AU - Chen, Jiehua

AU - Czerwiński, Wojciech

AU - Disser, Yann

AU - Feldmann, Andreas Emil

AU - Hermelin, Danny

AU - Nadara, Wojciech

AU - Pilipczuk, Marcin

AU - Pilipczuk, Michał

AU - Sorge, Manuel

AU - Wróblewski, Bartłomiej

AU - Zych-Pawlewicz, Anna

N1 - Funding Information:
∗Thiswork istheresultofresearchconductedwithinresearch project number 2017/26/D/ST6/00264 financed by National ScienceCentre(AnnaZych-Pawlewicz). AndreasEmilFeldmann was supportedby theCzech ScienceFoundation GAČR (grant#17-10090Y),andbytheCenter forFoundations ofModernComputer Science(CharlesUniv. projectUNCE/SCI/004). Thiswork isapartofprojectsthathavereceived fundingfromtheEuropeanResearchCouncil(ERC) undertheEuropeanUnion’sHorizon2020researchand innovation programme: GrantAgreements no.714704 (W.Nadara,Ma.Pilipczuk,M.Sorge)andno.677651 (Mi.Pilipczuk). Thefullversion ofthisextendedabstractcanbefoundat[13]. †TUWien,Austria,jiehua.chen@tuwien.ac.at. ‡UniversityofWarsaw, Poland, {wczerwin,w.nadara,marcin.pilipczuk, michal.pilipczuk, manuel.sorge, anka}@mimuw.edu.pl, bw371883@students.mimuw.edu.pl §TUDarmstadt,Germany, disser@mathematik.tu-darmstadt.de. ¶CharlesUniversityinPrague,Czechia, feldmann.a.e@gmail.com. ‖Ben-Gurion University of the Negev, Israel, hermelin@bgu.ac.il.
Funding Information:
This work is the result of research conducted within research project number 2017/26/D/ST6/00264 financed by National Science Centre (Anna Zych-Pawlewicz). Andreas Emil Feldmann was supported by the Czech Science Foundation GA?R (grant #17-10090Y), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004). This work is a part of projects that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme: Grant Agreements no. 714704 (W. Nadara, Ma. Pilipczuk, M. Sorge) and no. 677651 (Mi. Pilipczuk).
Publisher Copyright:
Copyright © 2021 by SIAM

PY - 2021/1/1

Y1 - 2021/1/1

N2 - We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time 2O(d2), which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time f(d) for some non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to dO(d). As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter k and a dynamic graph G, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: • Does G contain a simple path on k vertices? • Does G contain a simple cycle on at least k vertices? In the first case, the data structure achieves amortized update time 2O(k2). In the second case, the amortized update time is 2O(k4) + O(k log n). In both cases we assume access to a dictionary on the edges of G.

AB - We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time 2O(d2), which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time f(d) for some non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to dO(d). As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter k and a dynamic graph G, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: • Does G contain a simple path on k vertices? • Does G contain a simple cycle on at least k vertices? In the first case, the data structure achieves amortized update time 2O(k2). In the second case, the amortized update time is 2O(k4) + O(k log n). In both cases we assume access to a dictionary on the edges of G.

UR - http://www.scopus.com/inward/record.url?scp=85105334832&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85105334832

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 796

EP - 809

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

PB - Association for Computing Machinery

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

Y2 - 10 January 2021 through 13 January 2021

ER -