The Parameterized Complexity Of The K-Biclique Problem

Given a graph G and an integer k, the k-BICLIQUE problem asks whether G contains a complete bipartite subgraph with k vertices on each side. Whether there is an f(k).vertical bar G vertical bar(O)((1))-time algorithm, solving k-BICLIQUE for some computable function f has been a longstanding open problem.We show that k-BICLIQUE is W[1]-hard, which implies that such an f(k).vertical bar G vertical bar(O)((1))-time algorithm does not exist under the hypothesis W[1] not equal FPT from parameterized complexity theory. To prove this result, we give a reduction which, for every n-vertex graph G and small integer k, constructs a bipartite graph H = (L U R, E) in time polynomial in n such that if G contains a clique with k vertices. then there are k(k-1)/2 vertices in L with n(Theta)((1/k)) common neighbors; otherwise, any k(k-1)/2 vertices in L have at most (k+1)! common neighbors. An additional feature of this reduction is that it creates a gap on the right side of the biclique. Such a gap might have further applications in proving hardness of approximation results.Assuming a randomized version of Exponential Time Hypothesis, we establish an f(k).vertical bar G vertical bar(o(root K))-time lower bound for k-BICLIQUE for any computable function f. Combining our result with the work of Bulatov and Marx [2014], we obtain a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems.