Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

Authors R. Krithika , V. K. Kutty Malu, Roohani Sharma, Prafullkumar Tale

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Author Details

R. Krithika
  • Indian Institute of Technology Palakkad, India
V. K. Kutty Malu
  • Indian Institute of Technology Palakkad, India
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Prafullkumar Tale
  • Indian Institute of Science Education and Research Pune, India

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R. Krithika, V. K. Kutty Malu, Roohani Sharma, and Prafullkumar Tale. Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


A bipartite graph is called a biclique if it is a complete bipartite graph and a biclique is called a balanced biclique if it has equal number of vertices in both parts of its bipartition. In this work, we initiate the complexity study of Biclique Contraction and Balanced Biclique Contraction. In these problems, given as input a graph G and an integer k, the objective is to determine whether one can contract at most k edges in G to obtain a biclique and a balanced biclique, respectively. We first prove that these problems are NP-complete even when the input graph is bipartite. Next, we study the parameterized complexity of these problems and show that they admit single exponential-time FPT algorithms when parameterized by the number k of edge contractions. Then, we show that Balanced Biclique Contraction admits a quadratic vertex kernel while Biclique Contraction does not admit any polynomial compression (or kernel) unless NP ⊆ coNP/poly.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • contraction
  • bicliques
  • balanced bicliques
  • parameterized complexity


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