The Parameterized Complexity of the k-Biclique Problem

Given a graph <i>G</i> and an integer <i>k</i>, the <i>k</i>-B<scp>iclique</scp> problem asks whether <i>G</i> contains a complete bipartite subgraph with <i>k</i> vertices on each side. Whether there is an <i>f</i>(<i>k</i>) ċ |<i>G</i>|^<i>O</i>(1)-time algorithm, solving <i>k</i>-B<scp>iclique</scp> for some computable function <i>f</i> has been a longstanding open problem. We show that <i>k</i>-B<scp>iclique</scp> is <b>W[1]</b>-hard, which implies that such an <i>f</i>(<i>k</i>) ċ |<i>G</i>|^<i>O</i>(1)-time algorithm does not exist under the hypothesis <b>W[1]</b> ≠ <b>FPT</b> from parameterized complexity theory. To prove this result, we give a reduction which, for every <i>n</i>-vertex graph <i>G</i> and small integer <i>k</i>, constructs a bipartite graph <i>H</i> = (<i>L</i>⊍ <i>R</i>,<i>E</i>) in time polynomial in <i>n</i> such that if <i>G</i> contains a clique with <i>k</i> vertices, then there are <i>k</i>(<i>k</i> − 1)/2 vertices in <i>L</i> with <i>n</i>^{θ(1/<i>k</i>)} common neighbors; otherwise, any <i>k</i>(<i>k</i> − 1)/2 vertices in <i>L</i> have at most (<i>k</i>+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 <i>f</i>(<i>k</i>) ċ |<i>G</i>|^{<i>o</i>(&sqrt;<i>k</i>)}-time lower bound for <i>k</i>-<sc>Biclique</sc> for any computable function <i>f</i>. 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.