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.

翻译：本文首次对双团收缩与平衡双团收缩问题展开复杂性研究。在这两个问题中，给定输入图G和整数k，目标分别是判断能否通过至多k条边的收缩操作使G变为双团或平衡双团。首先证明即使输入图为二分图，这些问题仍是NP完全的。接着研究其参数化复杂性，证明当以边收缩数量k为参数时，这两个问题存在单指数时间的固定参数易处理算法。进一步表明，平衡双团收缩问题存在二次顶点核，而双团收缩问题在标准复杂性理论假设下不存在多项式压缩（或核）。