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 $K_{k,k}$. 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 $G=(A\;\dot\cup\;B, E)$ such that $A$ can be partitioned into $A=V_1\;\dot\cup \;V_2\;\dot\cup\cdots\dot\cup\; V_n$ and for every $s$ distinct indices $i_1\cdots i_s$, there exist $v_{i_1}\in V_{i_1}\cdots v_{i_s}\in V_{i_s}$ such that $v_{i_1}\cdots v_{i_s}$ 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 $B$. 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 $n^{O(1)}$. One corollary of our reduction is that there is no $f(k)\cdot n^{o(k)}$ time algorithm to decide whether a graph contains a subgraph isomorphic to $K_{k!,k!}$ unless the ETH(Exponential Time Hypothesis) fails. We also provide a probabilistic construction with better parameters $t=\Theta(s^2)$, which indicates that $k$-biclique has no $f(k)\cdot n^{o(\sqrt{k})}$-time algorithm unless 3-SAT with $m$ clauses can be solved in $2^{o(m)}$-time with high probability. Our result also implies the dichotomy classification of the parameterized complexity of cardinality constrain satisfaction problem and the inapproximability of maximum $k$-intersection problem.