research-article

Randomized Contractions Meet Lean Decompositions

Authors:

Daniel Lokshtanov,

Marcin Pilipczuk,

Michał Pilipczuk,

Magnus WahlströmAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 17, Issue 1

Article No.: 6, Pages 1 - 30

https://doi.org/10.1145/3426738

Published: 31 December 2020 Publication History

Abstract

We show an algorithm that, given an n-vertex graph G and a parameter k, in time 2^{O(k log k)} n^O(1) finds a tree decomposition of G with the following properties:

— every adhesion of the tree decomposition is of size at most k, and

— every bag of the tree decomposition is (i,i)-unbreakable in G for every 1 ⩽ i ⩽ k.

Here, a set X ⊆ V(G) is (a,b)-unbreakable in G if for every separation (A,B) of order at most b in G, we have |A \cap X| ⩽ a or |B ∩ X| ⩽ a.

The resulting tree decomposition has arguably best possible adhesion size bounds and unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for MINIMUM BISECTION, STEINER CUT, and STEINER MULTICUT with improved parameteric factor in the running time bound.

The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.

References

[1]

Noga Alon, Raphael Yuster, and Uri Zwick. 1995. Color-coding. J. ACM 42, 4 (1995), 844--856.

Digital Library

[2]

Patrick Bellenbaum and Reinhard Diestel. 2002. Two short proofs concerning tree-decompositions. Comb. Probab. Comput. 11, 6 (2002), 541--547.

Digital Library

[3]

Mikołaj Bojańczyk and Michal Pilipczuk. 2016. Definability equals recognizability for graphs of bounded treewidth. CoRR abs/1605.03045 (2016). arxiv:1605.03045 http://arxiv.org/abs/1605.03045 Extended abstract appeared in the proceedings of LICS 2016.

Digital Library

[4]

Karl Bringmann, Danny Hermelin, Matthias Mnich, and Erik Jan van Leeuwen. 2016. Parameterized complexity dichotomy for steiner Multicut. J. Comput. Syst. Sci. 82, 6 (2016), 1020--1043.

Digital Library

[5]

Jianer Chen, Yang Liu, Songjian Lu, Barry O’Sullivan, and Igor Razgon. 2008. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55, 5 (2008), 21:1--21:19.

Digital Library

[6]

Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michał Pilipczuk. 2016. Designing FPT algorithms for cut problems using randomized contractions. SIAM J. Comput. 45, 4 (2016), 1171--1229.

Digital Library

[7]

Rajesh Hemant Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michał Pilipczuk. 2012. Designing FPT algorithms for cut problems using randomized contractions. In 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12). IEEE Computer Society, 460--469.

Digital Library

[8]

Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Dániel Marx. 2013. Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. SIAM J. Comput. 42, 4 (2013), 1674--1696.

[9]

Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer.

Digital Library

[10]

Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. 2014. Minimum bisection is fixed parameter tractable. In 46th ACM Symposium on Theory of Computing (STOC’14). ACM, 323--332.

Digital Library

[11]

Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. 2013. On multiway cut parameterized above lower bounds. ACM Trans. Comput. Theory 5, 1 (2013), 3.

Digital Library

[12]

Serge Gaspers, Eun Jung Kim, Sebastian Ordyniak, Saket Saurabh, and Stefan Szeider. 2012. Don’t be strict in local search!. In 26th AAAI Conference on Artificial Intelligence (AAAI’12) AAAI Press. http://www.aaai.org/ocs/index.php/AAAI/AAAI12/paper/view/4929

Digital Library

[13]

Sylvain Guillemot. 2011. FPT algorithms for path-transversal and cycle-transversal problems. Discr. Opt. 8, 1 (2011), 61--71.

Digital Library

[14]

Yoichi Iwata. 2017. Linear-time kernelization for feedback vertex set. In 44th International Colloquium on Automata, Languages, and Programming (ICALP’17) (LIPIcs), Vol. 80. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, 68:1--68:14.

[15]

Yoichi Iwata, Magnus Wahlström, and Yuichi Yoshida. 2016. Half-integrality, lp-branching, and FPT algorithms. SIAM J. Comput. 45, 4 (2016), 1377--1411.

[16]

Yoichi Iwata, Yutaro Yamaguchi, and Yuichi Yoshida. 2017. Linear-time FPT algorithms via half-integral non-returning A-path packing. CoRR abs/1704.02700 (2017). arxiv:1704.02700 http://arxiv.org/abs/1704.02700

[17]

Ken-ichi Kawarabayashi and Mikkel Thorup. 2011. The minimum k-way cut of bounded size is fixed-parameter tractable. In 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’11). IEEE Computer Society, 160--169.

Digital Library

[18]

Stefan Kratsch and Magnus Wahlström. 2012. Representative sets and irrelevant vertices: New tools for kernelization. In 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12). IEEE Computer Society, 450--459.

Digital Library

[19]

Stefan Kratsch and Magnus Wahlström. 2014. Compression via matroids: A randomized polynomial kernel for odd cycle transversal. ACM Trans. Algorithms 10, 4 (2014), 20.

Digital Library

[20]

Daniel Lokshtanov and Dániel Marx. 2013. Clustering with local restrictions. Inf. Comput. 222 (2013), 278--292.

Digital Library

[21]

Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. 2018. Reducing CMSO model checking to highly connected graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP’18) (LIPIcs), Vol. 107. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, 135:1--135:14.

[22]

Dániel Marx. 2006. Parameterized graph separation problems. Theor. Comput. Sci. 351, 3 (2006), 394--406.

Digital Library

[23]

Dániel Marx, Barry O’Sullivan, and Igor Razgon. 2013. Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms 9, 4 (2013), 30:1--30:35.

Digital Library

[24]

Dániel Marx and Igor Razgon. 2014. Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput. 43, 2 (2014), 355--388.

[25]

Igor Razgon and Barry O’Sullivan. 2009. Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75, 8 (2009), 435--450.

Digital Library

[26]

Robin Thomas. 1990. A menger-like property of tree-width: The finite case. J. Comb. Theory, Ser. B 48, 1 (1990), 67--76.

Digital Library

Cited By

Gu H(2024)Rethinking the implementation and application of the Benczur-Karger minimum cuts algorithmInternational Journal of ADVANCED AND APPLIED SCIENCES10.21833/ijaas.2024.07.00711:7(57-62)Online publication date: Jul-2024
https://doi.org/10.21833/ijaas.2024.07.007
Kim EKratsch SPilipczuk MWahlström M(2024)Flow-augmentation II: Undirected GraphsACM Transactions on Algorithms10.1145/364110520:2(1-26)Online publication date: 13-Mar-2024
https://dl.acm.org/doi/10.1145/3641105
Anand ALee ELi JSaranurak TMohar BShinkar IO'Donnell R(2024)Approximating Small Sparse CutsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649747(319-330)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649747
Show More Cited By

Index Terms

Randomized Contractions Meet Lean Decompositions
1. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
    2. Parameterized complexity and exact algorithms
      1. Fixed parameter tractability

Recommendations

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

We introduce a new technique for designing fixed-parameter algorithms for cut problems, called randomized contractions. We apply our framework to obtain the first fixed-parameter algorithms (FPT algorithms) with exponential speed up for the Steiner Cut ...
Minimum bisection is fixed parameter tractable
STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

In the classic Minimum Bisection problem we are given as input a graph G and an integer k. The task is to determine whether there is a partition of V (G) into two parts A and B such that ||A| -- |B|| ≤ 1 and there are at most k edges with one endpoint ...
Clifford Algebras Meet Tree Decompositions

We introduce the non-commutative subset convolution--a convolution of functions useful when working with determinant-based algorithms. In order to compute it efficiently, we take advantage of Clifford algebras, a generalization of quaternions used ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 17, Issue 1

January 2021

335 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3446616

Editor:
Edith Cohen
Google Research, USA and Tel Aviv University, Israel

Issue’s Table of Contents

Copyright © 2020 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 December 2020

Accepted: 01 September 2020

Revised: 01 September 2020

Received: 01 October 2019

Published in TALG Volume 17, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

H2020 European Research Council
United States-Israel Binational Science Foundation
Narodowe Centrum Nauki
Swarnajayanti Fellowship

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

13
Total Citations
View Citations
212
Total Downloads

Downloads (Last 12 months)48
Downloads (Last 6 weeks)9

Reflects downloads up to 09 Aug 2024

Other Metrics

View Author Metrics

Citations

Cited By

Gu H(2024)Rethinking the implementation and application of the Benczur-Karger minimum cuts algorithmInternational Journal of ADVANCED AND APPLIED SCIENCES10.21833/ijaas.2024.07.00711:7(57-62)Online publication date: Jul-2024
https://doi.org/10.21833/ijaas.2024.07.007
Kim EKratsch SPilipczuk MWahlström M(2024)Flow-augmentation II: Undirected GraphsACM Transactions on Algorithms10.1145/364110520:2(1-26)Online publication date: 13-Mar-2024
https://dl.acm.org/doi/10.1145/3641105
Anand ALee ELi JSaranurak TMohar BShinkar IO'Donnell R(2024)Approximating Small Sparse CutsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649747(319-330)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649747
Kim EMasařík TPilipczuk MSharma RWahlström M(2024)On Weighted Graph Separation Problems and Flow AugmentationSIAM Journal on Discrete Mathematics10.1137/22M153118X38:1(170-189)Online publication date: 8-Jan-2024
https://doi.org/10.1137/22M153118X
Jana SSaha SSahu ASaurabh SVerma S(2024)Partitioning subclasses of chordal graphs with few deletionsTheoretical Computer Science10.1016/j.tcs.2023.114288983(114288)Online publication date: Feb-2024
https://doi.org/10.1016/j.tcs.2023.114288
Korhonen T(2023)A Single-Exponential Time 2-Approximation Algorithm for TreewidthSIAM Journal on Computing10.1137/22M147551X(FOCS21-174-FOCS21-194)Online publication date: 14-Nov-2023
https://doi.org/10.1137/22M147551X
Jana SSaha SSahu ASaurabh SVerma S(2023)Partitioning Subclasses of Chordal Graphs with Few DeletionsAlgorithms and Complexity10.1007/978-3-031-30448-4_21(293-307)Online publication date: 25-Apr-2023
https://doi.org/10.1007/978-3-031-30448-4_21
Lokshtanov DPilipczuk MPilipczuk MSaurabh SLeonardi SGupta A(2022)Fixed-parameter tractability of graph isomorphism in graphs with an excluded minorProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520076(914-923)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520076
Kim EKratsch SPilipczuk MWahlström MLeonardi SGupta A(2022)Directed flow-augmentationProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520018(938-947)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520018
Wahlström M(2022)Quasipolynomial Multicut-mimicking Networks and Kernels for Multiway Cut ProblemsACM Transactions on Algorithms10.1145/350130418:2(1-19)Online publication date: 4-Mar-2022
https://dl.acm.org/doi/10.1145/3501304
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents