Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

TL;DR

This work initiates a systematic study of contracting edges to obtain bicliques, focusing on Biclique Contraction and Balanced Biclique Contraction. It proves NP-hardness even on bipartite graphs, develops single-exponential FPT algorithms in the contraction budget $k$, and establishes a quadratic kernel for Balanced Biclique Contraction while proving that Biclique Contraction admits no polynomial kernel under standard complexity assumptions. The paper further analyzes the broader family ${K_{p,*}}$-Contraction, showing ${ m NP}$-hardness for fixed $p\ge 2$ via PNAESAT and providing $O^*(3^k)$-time FPT algorithms for fixed $p$, with ${K_{1,*}}$-Contraction covered by existing results. These results advance the understanding of graph-contraction problems, offering practical fixed-parameter strategies and precise kernelization limits for targeted biclique outcomes.

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.

Figures (3)

• Figure 1: The graph $H$ in the reduction from Red Blue Dominating Set to Biclique Contraction where edges between $R$ and $B$ are the same as in $G$. The vertex $x$ is adjacent to every vertex in $R \cup C$, and each vertex in $B$ is adjacent to exactly one vertex in $B'$.
• Figure 2: Construction of graph $H$ in the reduction from Hypergraph 2-Coloring to Balanced Biclique Contraction where thick lines between two sets indicate bicliques.
• Figure 3: The graph $G$ corresponding to the formula $\phi$ in the reduction from PNAESAT to $K_{p, *}$-Contraction. A thick line between two sets of vertices denotes that every vertex of one set is adjacent to every vertex of the other set.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• Definition 7: $k$-Constrained Valid Partition
• Lemma 8
• proof