The Parameterized Complexity of k-Biclique

Given a graph G and a parameter k, the k-biclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is the most easily stated problem on graphs whose parameterized complexity is still unknown. We provide an fpt-reduction from k-clique to k-biclique, thus solving this longstanding open problem. Our reduction use a class of bipartite graphs with a threshold property of independent interest. More specifically, for positive integers n, s and t, we consider a bipartite graph G = (A ∪̇ B,E) such that A can be partitioned into A = V1 ∪̇ V2 ∪̇ · · · ∪̇ Vn and for every s distinct indices i1 · · · is, there exist vi1 ∈ Vi1 · · · vis ∈ Vis such that vi1 · · · vis have at least t + 1 common neighbors in B; on the other hand, every s + 1 distinct vertices in A have at most t common neighbors in B. We prove that given such threshold bipartite graphs, we can construct an fpt-reduction from the k-clique problem to the k-biclique problem. Using the Paley-type graphs and Weil’s character sum theorem, we show that for t = (s + 1)! and n large enough, such threshold bipartite graphs can be computed in n. One corollary of our reduction is that there is no f(k) · n time algorithm to decide whether a graph contains a subgraph isomorphic to Kk!,k! unless the ETH(Exponential Time Hypothesis) fails. We also provide a probabilistic construction with better parameters t = Θ(s), which indicates that k-biclique has no f(k) · n √ -time algorithm unless 3-SAT with m clauses can be solved in 2-time with high probability. Besides the lower bound for exact computation of k-biclique, our result also implies the dichotomy classification of the parameterized complexity of cardinality constrain satisfaction problem and the inapproximability of maximum k-intersection problem.