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.

翻译：本研究首次对二部图收缩与平衡二部图收缩问题展开复杂度分析。在这两类问题中，给定图G与整数k作为输入，目标分别是判断能否通过至多k次边收缩操作将G转化为二部图与平衡二部图。我们首先证明了即使输入图是二部图，这些问题仍是NP完全问题。随后，我们研究了这些问题的参数化复杂度，证明了在以边收缩次数k为参数时，它们存在单指数时间复杂度的FPT算法。进一步研究表明，平衡二部图收缩问题具有二次规模的顶点核，而基于标准计算复杂性理论假设，二部图收缩问题不存在任何多项式规模的压缩（或核）。我们还针对受限二部图的收缩问题提出了更快速的FPT算法。