Given a bipartite graph, the maximum balanced biclique (\textsf{MBB}) problem, discovering a mutually connected while equal-sized disjoint sets with the maximum cardinality, plays a significant role for mining the bipartite graph and has numerous applications. Despite the NP-hardness of the \textsf{MBB} problem, in this paper, we show that an exact \textsf{MBB} can be discovered extremely fast in bipartite graphs for real applications. We propose two exact algorithms dedicated for dense and sparse bipartite graphs respectively. For dense bipartite graphs, an $\mathcal{O}^{*}( 1.3803^{n})$ algorithm is proposed. This algorithm in fact can find an \textsf{MBB} in near polynomial time for dense bipartite graphs that are common for applications such as VLSI design. This is because, using our proposed novel techniques, the search can fast converge to sufficiently dense bipartite graphs which we prove to be polynomially solvable. For large sparse bipartite graphs typical for applications such as biological data analysis, an $\mathcal{O}^{*}( 1.3803^{\ddotδ})$ algorithm is proposed, where $\ddotδ$ is only a few hundreds for large sparse bipartite graphs with millions of vertices. The indispensible optimizations that lead to this time complexity are: we transform a large sparse bipartite graph into a limited number of dense subgraphs with size up to $\ddotδ$ and then apply our proposed algorithm for dense bipartite graphs on each of the subgraphs. To further speed up this algorithm, tighter upper bounds, faster heuristics and effective reductions are proposed, allowing an \textsf{MBB} to be discovered within a few seconds for bipartite graphs with millions of vertices. Extensive experiments are conducted on synthetic and real large bipartite graphs to demonstrate the efficiency and effectiveness of our proposed algorithms and techniques.
Identifying dense bipartite subgraphs is a common graph data mining task. Many applications focus on the enumeration of all maximal bicliques (MBs), though sometimes the stricter variant of maximal induced bicliques (MIBs) is of interest. Recent work of Kloster et al. introduced a MIB-enumeration approach designed for "near-bipartite" graphs, where the runtime is parameterized by the size k of an odd cycle transversal (OCT), a vertex set whose deletion results in a bipartite graph. Their algorithm was shown to outperform the previously best known algorithm even when k was logarithmic in |V|. I…
Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction
Indian Institute of Technology Palakkad（印度理工学院帕拉卡德分校）；Institute for Basic Science（基础科学研究院）；Indian Institute of Science Education and Research Pune（印度科学教育与研究学院浦那分校）
Maximal biclique enumeration is a fundamental problem in bipartite graph data analysis. Existing biclique enumeration methods mainly focus on non-attributed bipartite graphs and also ignore the \emph{fairness} of graph attributes. In this paper, we introduce the concept of fairness into the biclique model for the first time and study the problem of fairness-aware biclique enumeration. Specifically, we propose two fairness-aware biclique models, called \nonesidebc~and \ntwosidebc~respectively. To efficiently enumerate all {\nonesidebc}s, we first present two non-trivial pruning techniques, call…
We consider the maintenance of maximal bicliques from a dynamic bipartite graph that changes over time due to the addition or deletion of edges. When the set of edges in a graph changes, we are interested in knowing the change in the set of maximal bicliques (the "change"), rather than in knowing the set of maximal bicliques that remain unaffected. The challenge in an efficient algorithm is to enumerate the change without explicitly enumerating the set of all maximal bicliques. In this work, we present (1) near-tight bounds on the magnitude of change in the set of maximal bicliques of a graph,…
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 just identifying 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 fas…
A 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic if and only if there is a clique of size $n$ in the weak modular product between the two graphs. Restricting to bipartite graphs and considering complete bipartite subgraphs (bicliques) therein, we study the combinatorics of the weak modular product. We identify cases where isomorphism is tractable using this approach, which we call Isomorphism via Biclique Enumeration (IvBE). We find that IvBE is polynomial for bipartite $2K_2$-free graphs and quasi-polynomial for families of bipartite graphs, where the largest indu…
Detecting dense subgraphs from large graphs is a core component in many applications, ranging from social networks mining, bioinformatics. In this paper, we focus on mining dense subgraphs in a bipartite graph. The work is motivated by the task of detecting synchronized behavior that can often be formulated as mining a bipartite graph formed by the source nodes (followers, customers) and target nodes (followees, products, etc.) for malicious patterns. We introduce a new restricted biclique problem, Maximal Half Isolated Biclique (MHI Biclique), and show that the problem finds immediate applica…
National Research University Higher School of Economics（俄罗斯国立高等经济研究大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
This paper is related to the problem of finding the maximal quasi-bicliques in a bipartite graph (bigraph). A quasi-biclique in the bigraph is its "almost" complete subgraph. The relaxation of completeness can be understood variously; here, we assume that the subgraph is a $γ$-quasi-biclique if it lacks a certain number of edges to form a biclique such that its density is at least $γ\in (0,1]$. For a bigraph and fixed $γ$, the problem of searching for the maximal quasi-biclique consists of finding a subset of vertices of the bigraph such that the induced subgraph is a quasi-biclique and its si…
A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander $Δ$-regular graphs, including the canonical class of bipartite random $Δ$-regular graphs. …
Maximal Biclique Enumeration with Improved Worst-Case Time Complexity Guarantee: A Partition-Oriented Strategy
College of Computer Science, Beijing University of Technology（北京工业大学计算机学院）；State Key Laboratory of Novel Software Technology, Nanjing University（南京大学软件新技术国家重点实验室）；College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）
The problem of finding the largest induced balanced bipartite subgraph in a given graph is NP-hard. This problem is closely related to the problem of finding the smallest Odd Cycle Transversal. In this work, we consider the following model of instances: starting with a set of vertices $V$, a set $S \subseteq V$ of $k$ vertices is chosen and an arbitrary $d$-regular bipartite graph is added on it; edges between pairs of vertices in $S \times (V \setminus S)$ and $(V \setminus S) \times (V \setminus S)$ are added with probability $p$. Since for $d=0$, the problem reduces to recovering a planted …
We study the following general disjoint paths problem: given a supply graph $G$, a set $T\subseteq V(G)$ of terminals, a demand graph $H$ on the vertices $T$, and an integer $k$, the task is to find a set of $k$ pairwise vertex-disjoint valid paths, where we say that a path of the supply graph $G$ is valid if its endpoints are in $T$ and adjacent in the demand graph $H$. For a class $\mathcal{H}$ of graphs, we denote by $\mathcal{H}$-Maximum Disjoint Paths the restriction of this problem when the demand graph $H$ is assumed to be a member of $\mathcal{H}$. We study the fixed-parameter tractabi…
We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, List $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving NP-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained through a proof that the Surjective $C_6$-Homomorphism problem is NP-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vika…
University of Essex（埃塞克斯大学）；University of Nottingham（诺丁汉大学）；Simon Fraser University（西蒙菲莎大学）
We study the Bipartite Boolean Quadratic Programming Problem (BBQP) which is an extension of the well known Boolean Quadratic Programming Problem (BQP). Applications of the BBQP include mining discrete patterns from binary data, approximating matrices by rank-one binary matrices, computing the cut-norm of a matrix, and solving optimisation problems such as maximum weight biclique, bipartite maximum weight cut, maximum weight induced sub-graph of a bipartite graph, etc. For the BBQP, we first present several algorithmic components, specifically, hill climbers and mutations, and then show how to…
We study the fundamental problem of butterfly (i.e. (2,2)-bicliques) counting in bipartite streaming graphs. Similar to triangles in unipartite graphs, enumerating butterflies is crucial in understanding the structure of bipartite graphs. This benefits many applications where studying the cohesion in a graph shaped data is of particular interest. Examples include investigating the structure of computational graphs or input graphs to the algorithms, as well as dynamic phenomena and analytic tasks over complex real graphs. Butterfly counting is computationally expensive, and known techniques do …
A bicolored rectangular family BRF is a collection of all axis-parallel rectangles contained in a given region Z of the plane formed by selecting a bottom-left corner from a set A and an upper-right corner from a set B. We prove that the maximum independent set and the minimum hitting set of a BRF have the same cardinality and devise polynomial time algorithms to compute both. As a direct consequence, we obtain the first polynomial time algorithm to compute minimum biclique covers, maximum cross-free matchings and jump numbers in a class of bipartite graphs that significantly extends convex bi…
The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph $G$ whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of $G$ having the same number of blue vertices and red vertices. They showed that this problem is NP-hard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in $O(n^3)$ time for trees and $O(n + m)$ time for split graphs and properly colored bipartite graphs, where $n$ is the number of vertices and…
