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An exact exponential time algorithm for counting bipartite cliques
"... We present a simple exact algorithm for counting bicliques of given size in a bipartite graph on n vertices. We achieve running time of O(1.2491 n), improving upon known exact algorithms for finding and counting bipartite cliques. ..."
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We present a simple exact algorithm for counting bicliques of given size in a bipartite graph on n vertices. We achieve running time of O(1.2491 n), improving upon known exact algorithms for finding and counting bipartite cliques.
The parameterized complexity of kBiclique
 In Proc. 26th SODA
, 2014
"... Given a graph G and a parameter k, the kBiclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that kBiclique is W[1]hard by giving an fptreduction from k ..."
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Given a graph G and a parameter k, the kBiclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that kBiclique is W[1]hard by giving an fptreduction from kClique to kBiclique, thus solving this longstanding open problem. Our reduction uses a class of bipartite graphs with a certain threshold property, which might be of some independent interest. More precisely, 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 fptreduction from kClique to kBiclique. Using the Paleytype 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 polynomial time. One corollary of our reduction is that there is no f(k) ·no(k) time algorithm to decide whether a graph contains a subgraph isomorphic to Kk!,k! unless the Exponential Time Hypothesis (ETH) fails. We also provide a probabilistic construction with better parameters t = Θ(s2), which indicates that kBiclique has no f(k) · no( k)time algorithm unless 3SAT with m clauses can be solved in 2o(m)time with high probability. Besides the lower bound for exact computation of kBiclique, our result also implies a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems and the inapproximability of the maximum kintersection problem.
Hereditary bicliqueHelly graphs: recognition and maximal biclique enumeration. arXiv
"... A biclique is a set of vertices that induce a bipartite complete graph. A graph G is bicliqueHelly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also bicliqueHelly, then G is hereditary bicliqueHelly. A graph is C4dominated when every cycle ..."
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A biclique is a set of vertices that induce a bipartite complete graph. A graph G is bicliqueHelly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also bicliqueHelly, then G is hereditary bicliqueHelly. A graph is C4dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary bicliqueHelly graphs is formed precisely by those C4dominated graphs that contain no triangles and no induced cycles of length either 5, or 6. Using this characterization, we develop an algorithm for recognizing hereditary bicliqueHelly graphs in O(n2 + αm) time and O(m) space. (Here n, m, and α = O(m1/2) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4dominated graphs that contain no triangles in O(αm) time and O(m) space. Finally, we show how to enumerate all the maximal bicliques of a C4dominated graph with no triangles in O(n 2 + αm) time and O(αm) space, and we discuss how some biclique problems can be solved in O(αm) time and O(n+m) space.