research-article

Approximating Pathwidth for Graphs of Small Treewidth

Authors:

Carla Groenland,

Gwenaël Joret,

Wojciech Nadara,

Bartosz WalczakAuthors Info & Claims

ACM Transactions on Algorithms, Volume 19, Issue 2

Article No.: 16, Pages 1 - 19

https://doi.org/10.1145/3576044

Published: 09 March 2023 Publication History

Abstract

We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of \(O(t\sqrt {\log t})\). This is the first algorithm to achieve an f(t)-approximation for some function f.

Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA’18): there exists a universal constant c such that every graph with pathwidth Ω(k^c) has treewidth at least k or contains a subdivision of a complete binary tree of height k.

Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t′ in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC’05) for treewidth.

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Cited By

Hecher MKiesel RWooldridge MDy JNatarajan S(2024)On the structural hardness of answer set programmingProceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v38i9.28923(10535-10543)Online publication date: 20-Feb-2024
https://dl.acm.org/doi/10.1609/aaai.v38i9.28923
Bodlaender H(2024)Approximation Algorithms for Treewidth, Pathwidth, and Treedepth—A Short SurveyGraph-Theoretic Concepts in Computer Science10.1007/978-3-031-75409-8_1(3-18)Online publication date: 19-Jun-2024
https://dl.acm.org/doi/10.1007/978-3-031-75409-8_1
Hatzel MJoret GMicek PPilipczuk MUeckerdt TWalczak B(2023)Tight Bound on Treedepth in Terms of Pathwidth and Longest PathCombinatorica10.1007/s00493-023-00077-w44:2(417-427)Online publication date: 19-Dec-2023
https://dl.acm.org/doi/10.1007/s00493-023-00077-w

Index Terms

Approximating Pathwidth for Graphs of Small Treewidth
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Approximation algorithms
      2. Graph algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 19, Issue 2

April 2023

367 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3582899

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

Issue’s Table of Contents

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 the author(s) 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].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 March 2023

Online AM: 13 December 2022

Accepted: 30 November 2022

Revised: 24 November 2022

Received: 29 March 2021

Published in TALG Volume 19, Issue 2

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Polish National Agency for Academic Exchange (NAWA)

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Cited By

Hecher MKiesel RWooldridge MDy JNatarajan S(2024)On the structural hardness of answer set programmingProceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence and Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence and Fourteenth Symposium on Educational Advances in Artificial Intelligence10.1609/aaai.v38i9.28923(10535-10543)Online publication date: 20-Feb-2024
https://dl.acm.org/doi/10.1609/aaai.v38i9.28923
Bodlaender H(2024)Approximation Algorithms for Treewidth, Pathwidth, and Treedepth—A Short SurveyGraph-Theoretic Concepts in Computer Science10.1007/978-3-031-75409-8_1(3-18)Online publication date: 19-Jun-2024
https://dl.acm.org/doi/10.1007/978-3-031-75409-8_1
Hatzel MJoret GMicek PPilipczuk MUeckerdt TWalczak B(2023)Tight Bound on Treedepth in Terms of Pathwidth and Longest PathCombinatorica10.1007/s00493-023-00077-w44:2(417-427)Online publication date: 19-Dec-2023
https://dl.acm.org/doi/10.1007/s00493-023-00077-w

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