Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

TL;DR

This paper explores the computational complexity of contracting edges in graphs to form bicliques and balanced bicliques, establishing NP-completeness, fixed-parameter tractability, kernelization results, and improved algorithms.

Contribution

It provides the first complexity analysis of Biclique Contraction and Balanced Biclique Contraction, including NP-completeness, FPT algorithms, kernelization bounds, and faster algorithms for special cases.

Findings

NP-complete even for bipartite graphs

FPT algorithms with single exponential time

Quadratic kernel for Balanced Biclique Contraction

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.