research-article

Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems

Authors:

Michał Pilipczuk,

Lukáš MachAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 16, Issue 2

Article No.: 25, Pages 1 - 31

https://doi.org/10.1145/3381426

Published: 11 March 2020 Publication History

Abstract

In this work, we focus on several completion problems for subclasses of chordal graphs: MINIMUM FILL-IN, INTERVAL COMPLETION, PROPER INTERVAL COMPLETION, TRIVIALLY PERFECT COMPLETION, and THRESHOLD COMPLETION. In these problems, the task is to add at most k edges to a given graph to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related CHAIN COMPLETION problem:

• Assuming the Exponential Time Hypothesis, none of these problems can be solved in time 2^O(n^1/2/log^c n) or 2^O(k^1/4/log^c k)· n^O(1), for some integer c.

• Assuming the non-existence of a subexponential-time approximation scheme for MIN BISECTION on d-regular graphs, for some constant d, none of these problems can be solved in time 2^o(n) or 2^o√k)}· n^O(1).

For all the aforementioned completion problems, apart from PROPER INTERVAL COMPLETION, FPT algorithms with running time of the form 2^{O(√k log k)}· n^O(1) are known. Thus, the second result proves that a significant improvement of any of these algorithms would lead to a surprising breakthrough in the design of approximation algorithms for MIN BISECTION.

To prove our results, we use a reduction methodology based on combining the classic approach of starting with a sparse instance of 3-SAT, prepared using the Sparsification Lemma, with the existence of almost linear-size Probabilistically Checkable Proofs. Apart from our main results, we also obtain lower bounds excluding the existence of subexponential algorithms for the OPTIMUM LINEAR ARRANGEMENT problem, as well as improved, yet still not tight, lower bounds for FEEDBACK ARC SET IN TOURNAMENTS.

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Cited By

Włodarczyk M(2025)Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by TreewidthAlgorithmica10.1007/s00453-025-01293-0Online publication date: 29-Jan-2025
https://doi.org/10.1007/s00453-025-01293-0
Crespelle CDrange PFomin FGolovach P(2023)A survey of parameterized algorithms and the complexity of edge modificationComputer Science Review10.1016/j.cosrev.2023.10055648(100556)Online publication date: May-2023
https://doi.org/10.1016/j.cosrev.2023.100556
Crespelle CGras BPerez A(2021)Completion to Chordal Distance-Hereditary Graphs: A Quartic Vertex-KernelGraph-Theoretic Concepts in Computer Science10.1007/978-3-030-86838-3_12(156-168)Online publication date: 23-Jun-2021
https://dl.acm.org/doi/10.1007/978-3-030-86838-3_12

Index Terms

Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis
    2. Parameterized complexity and exact algorithms
      1. Fixed parameter tractability

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Published In

ACM Transactions on Algorithms Volume 16, Issue 2

April 2020

372 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3386689

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2020 ACM.

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Publication History

Published: 11 March 2020

Accepted: 01 December 2019

Revised: 01 July 2019

Received: 01 May 2016

Published in TALG Volume 16, Issue 2

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Graph completion problems

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European Research Council's
Government of the Russian Federation
Warsaw Center of Mathematics and Computer Science
Foundation for Polish Science via the START stipend programme
Polish National Science Center
European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC

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Cited By

Włodarczyk M(2025)Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by TreewidthAlgorithmica10.1007/s00453-025-01293-0Online publication date: 29-Jan-2025
https://doi.org/10.1007/s00453-025-01293-0
Crespelle CDrange PFomin FGolovach P(2023)A survey of parameterized algorithms and the complexity of edge modificationComputer Science Review10.1016/j.cosrev.2023.10055648(100556)Online publication date: May-2023
https://doi.org/10.1016/j.cosrev.2023.100556
Crespelle CGras BPerez A(2021)Completion to Chordal Distance-Hereditary Graphs: A Quartic Vertex-KernelGraph-Theoretic Concepts in Computer Science10.1007/978-3-030-86838-3_12(156-168)Online publication date: 23-Jun-2021
https://dl.acm.org/doi/10.1007/978-3-030-86838-3_12

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