Finding Maximum Edge Bicliques in Convex Bipartite Graphs

Abstract

A bipartite graph G = (A, B, E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v ∈ A, vertices adjacent to v are consecutive in B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G = (A, B, E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(n log 3 n log log n) time and O(n) space, where n = |A|. This improves the current O(n 2 ) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely 312 Algorithmica (2012) 64:311-325 for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(nα(n)) and O(n) time, respectively, where n = min(|A|, |B|) and α(n) is the slowly growing inverse of the Ackermann function.

Polynomial-time algorithms for Subgraph Isomorphism in small graph classes of perfect graphs

“…An interesting problem left unsettled is the complexity of Subgraph Isomorphism where the base graphs are bipartite permutation graphs and the pattern graphs are chain graphs. It is known that although the maximum edge biclique problem is NP-complete for general bipartite graphs [28], it can be solved in polynomial time for some super classes of bipartite permutation graphs (see [26]). Therefore, it might be possible to have a polynomial-time algorithm for Subgraph Isomorphism when the pattern graphs are chain graphs and the base graphs belong to an even larger class like convex graphs.…”

On finding the maximum edge biclique in a bipartite graph: a subspace clustering approach

“…A wealth of research has been undertaken to study the maximum edge biclique problem. This research has received much attention due to its wide range of applications in areas such as bioinformatics [22], epidemiology [17], formal concept analysis [6], manufacturing problems [3], molecular biology [18], machine learning [15], management science [23], and conjunctive clustering [15].…”

“…Among the algorithms in the first category we can find: bounding the biclique's size [22], bounding the graph's arboricity [4], and bounding the graph's vertices degree [1]. Among the algorithms in the second category we can find: limitation to convex bipartite graphs [1,7,18], and limitation to chordal bipartite graphs [7]. Among the algorithms in the last category we can find: reduction to maximal clique [12,24], and reduction to frequent itemsets [10,25,27].…”

“…Biclique detection is a well-known problem in graph theory and data mining, with numerous real-world applications across different domains [3,6,9,13,15,17,18,22,23,29]. Take for example text networks depicting the relationship between words and documents.…”

A GRASP algorithm with Tabu Search improvement for solving the maximum intersection of k-subsets problem

“…Thus, a company can publish a set of attributes if and only if, at least, k people have the same attributes in common. Another relevant application emerges in the context of genetics, where it is required to identify k genes with the maximum number of features in common to conform DNA Microarray (see (Nussbaum et al 2010) for further details). Finally, in Bogue et al (2014) a novel application related with recommender systems is presented.…”