Mining Maximal Induced Bicliques using Odd Cycle Transversals

Abstract

Many common graph data mining tasks take the form of identifying dense subgraphs (e.g. clustering, clique-finding, etc). In biological applications, the natural model for these dense substructures is often a complete bipartite graph (biclique), and the problem requires enumerating all maximal bicliques (instead of identifying just the largest or densest). The best known algorithm in general graphs is due to Dias et al., and runs in time O(M |V | 4 ), where M is the number of maximal induced bicliques (MIBs) in the graph. When the graph being searched is itself bipartite, Zhang et al. give a faster algorithm where the time per MIB depends on the number of edges in the graph. In this work, we present a new algorithm for enumerating MIBs in general graphs, whose run time depends on how "close to bipartite" the input is. Specifically, the runtime is parameterized by the size k of an odd cycle transversal (OCT), a vertex set whose deletion results in a bipartite graph. Our algorithm runs in time O(M |V ||E|k 2 3 k/3 ), which is an improvement on Dias et al. whenever k ≤ 3 log 3 |V |. We implement our algorithm alongside a variant of Dias et al.'s in open-source C++ code, and experimentally verify that the OCT-based approach is faster in practice on graphs with a wide variety of sizes, densities, and OCT decompositions.

Efficient Mining of Maximal Bicliques in Graph by Pruning Search Space

“…If an MB consists of component sets at least one of which is not an independent vertex set, the MB is not enumerated. Kloster et al [10] pursued improving the algorithm of Dias et al But their algorithm is specifically designed for general graphs which are near to bipartite graphs. Their algorithm has time complexity of O(knmh 2 3 h/3 ) where h is the cardinality of the vertex set whose deletion from G makes G a bipartite graph.…”

Faster Biclique Mining in Near-Bipartite Graphs

“…We began by running our algorithms on the same corpus of graphs as in [17] (see 5.2). As the new algorithms finished considerably faster than those in [17], we were able to scale up both n B and n O to create new sets of experiments, discussed in 5.3. We also ran our algorithms on computational biology graphs from [29], which have been shown to be near-bipartite; these results are in 5.4.…”

“…We begin by evaluating our algorithms on the corpus of graphs used in [17]. This dataset was designed to independently test the effect of each parameter (the ex-pected densities in various regions of the graph, the cv values, n O , n B , and n L /n R ) on the algorithms' runtime.…”

“…Thus OCT-MICA is able to scale to graphs with considerably larger OCT sets and higher density than both MICA and the MIB-enumerating algorithms. We additionally created graphs with n O > 3 log 3 n B , which was not done in [17], and ran the algorithms for both MIBs and MBs (Figure 4). These graphs had n B values up to 4000 and for each value of n B , we used three values of n O ; 10, 3 log 3 n B , and √ n B .…”

“…Until recently, the only known non-trivial algorithm for enumerating maximal induced bicliques (MIBs) in general graphs was that of Dias et al which did so in lexicographic order [5]. In [17], Kloster . In contrast to OCT-MIB, the worst-case runtime of OCT-MIB-II is not dependent on the number of MISs in O, making it better than OCT-MIB when n O ∈ ω (1).…”