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arXiv · 2307.10607

Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

R. Krithika·V. K. Kutty Malu·Roohani Sharma·Prafullkumar Tale
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Abstract

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) under standard complexity-theoretic assumptions. We also give faster FPT algorithms for contraction to restricted bicliques.

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R. Krithika, V. K. Kutty Malu, Roohani Sharma, Prafullkumar Tale. 2023-07-20. Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction. https://arxiv.org/abs/2307.10607

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