research-article

The Parameterized Complexity of the k-Biclique Problem

Author:

Bingkai LinAuthors Info & Claims

Journal of the ACM (JACM), Volume 65, Issue 5

Article No.: 34, Pages 1 - 23

https://doi.org/10.1145/3212622

Published: 29 August 2018 Publication History

Abstract

Given a graph G and an integer k, the k-Biclique problem asks whether G contains a complete bipartite subgraph with k vertices on each side. Whether there is an f(k) ċ |G|^O(1)-time algorithm, solving k-Biclique for some computable function f has been a longstanding open problem.

We show that k-Biclique is W[1]-hard, which implies that such an f(k) ċ |G|^O(1)-time algorithm does not exist under the hypothesis W[1] ≠ FPT from parameterized complexity theory. To prove this result, we give a reduction which, for every n-vertex graph G and small integer k, constructs a bipartite graph H = (L⊍ R,E) in time polynomial in n such that if G contains a clique with k vertices, then there are k(k − 1)/2 vertices in L with n^θ(1/k) common neighbors; otherwise, any k(k − 1)/2 vertices in L have at most (k+1)! common neighbors. An additional feature of this reduction is that it creates a gap on the right side of the biclique. Such a gap might have further applications in proving hardness of approximation results.

Assuming a randomized version of Exponential Time Hypothesis, we establish an f(k) ċ |G|^o(√k)-time lower bound for k-Biclique for any computable function f. Combining our result with the work of Bulatov and Marx [2014], we obtain a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems.

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Deligkas AEiben EGoldsmith TDastani MSichman JAlechina NDignum V(2024)The Parameterized Complexity of Welfare Guarantees in Schelling SegregationProceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems10.5555/3635637.3662892(425-433)Online publication date: 6-May-2024
https://dl.acm.org/doi/10.5555/3635637.3662892
Guruswami VLin BRen XSun YWu KMohar BShinkar IO'Donnell R(2024)Parameterized Inapproximability Hypothesis under Exponential Time HypothesisProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649771(24-35)Online publication date: 10-Jun-2024
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C.M. Gomes GLegrand-Duchesne CMahmoud RMouawad AOkamoto YF. dos Santos Vvan der Zanden T(2024)Minimum separator reconfigurationJournal of Computer and System Sciences10.1016/j.jcss.2024.103574146(103574)Online publication date: Dec-2024
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Index Terms

The Parameterized Complexity of the k-Biclique Problem
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Extremal graph theory
      2. Random graphs
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Design and analysis of algorithms
    1. Parameterized complexity and exact algorithms
      1. W hierarchy

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 65, Issue 5

October 2018

299 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3274534

Editor:
Éva Tardos
Cornell University

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Copyright © 2018 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 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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Publication History

Published: 29 August 2018

Accepted: 01 April 2018

Revised: 01 October 2017

Received: 01 October 2016

Published in JACM Volume 65, Issue 5

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https://dl.acm.org/doi/10.5555/3635637.3662892
Guruswami VLin BRen XSun YWu KMohar BShinkar IO'Donnell R(2024)Parameterized Inapproximability Hypothesis under Exponential Time HypothesisProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649771(24-35)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649771
C.M. Gomes GLegrand-Duchesne CMahmoud RMouawad AOkamoto YF. dos Santos Vvan der Zanden T(2024)Minimum separator reconfigurationJournal of Computer and System Sciences10.1016/j.jcss.2024.103574146(103574)Online publication date: Dec-2024
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