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In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with the uniform distribution of all possible edge $k$-colorings of $G$. A special case of our results is the solution space of the possible completions of Latin rectangles., Comment: 20 pages, 2 figures

Consensus-based data aggregation in d-regular bipartite graphs poses a challenging task for the scientific community since some of these algorithms diverge in this critical graph topology. Nevertheless, one can see a lack of scientific studies dealing with this topic in the literature. Motivated by our recent research concerned with this issue, we provide a comparative study of frequently applied consensus algorithms for distributed averaging in d-regular bipartite graphs in this paper. More specifically, we examine the performance of these algorithms with bounded execution in this topology in order to identify which algorithm can achieve the consensus despite no reconfiguration and find the best-performing algorithm in these graphs. In the experimental part, we apply the number of iterations required for consensus to evaluate the performance of the algorithms in randomly generated regular bipartite graphs with various connectivities and for three configurations of the applied stopping criterion, allowing us to identify the optimal distributed consensus algorithm for this graph topology. Moreover, the obtained experimental results presented in this paper are compared to other scientific manuscripts where the analyzed algorithms are examined in non-regular non-bipartite topologies.

The problem of maximum coupling in bipartite weighted graphs has important practical applications in defense and security. For instance, in the context of border security, the problem arises in the allocation of resources to monitor and respond to potential threats, where the weights on the edges represent the effectiveness of different resources for detecting or mitigating specific threats. Finding the maximum coupling in such a graph can ensure that the most effective resources are allocated to the most critical areas, thereby improving border security.In addition to military applications, the problem of maximum coupling in bipartite weighted graphs can also enhance network security. By modeling a network as a bipartite graph with nodes representing network components and edges representing their connections, the maximum coupling search method can be used to identify potential security breaches. Any unmatched node in the graph suggests that the component may be compromised or vulnerable to attack, making the maximum coupling search method an effective tool for prioritizing and improving network security. This paper presents a new method for efficiently finding the maximum coupling in a bipartite weighted graph. The proposed method employs an iterative approach that builds on the traditional augmenting path algorithm, but incorporates additional heuristics and pruning techniques to reduce computational complexity. Experimental results demonstrate that the proposed method outperforms existing algorithms in terms of both accuracy and speed. The method's effectiveness and efficiency make it a valuable tool for enhancing defense and security in various applications, including military operations and network security.

This paper considers a bipartite graph where a perfect matching does not necessarily exist. Linear programming is used in this paper as a special case of the linear program is the assignment problem, which is another name for the weighted maximum matching problem. The objective is to show that linear programming, in particular the simplex algorithm, can be used to calculate maximum weight matchings and minimum weighted vertex covers. This study also calculates and shows the equivalence of the maximum matching and minimum vertex cover cardinalities, and uses linear programming duality to present its relevance to König's theorem. Finally, Hall’s marriage theorem is used to explicitly prove the absence of a perfect matching in particular graphs. There exist many algorithms developed over the years that are designed to solve maximum matching problems in polynomial time, but the simplex method is one of the oldest and simplest algorithms to understand in solving these problems.

Let [Formula: see text] be a bipartite graph over the vertex set [Formula: see text] and let [Formula: see text] be the cover ideal of [Formula: see text] in the polynomial ring [Formula: see text]. It is known that there are integers [Formula: see text] and [Formula: see text] such that [Formula: see text] is a linear function in [Formula: see text] for all [Formula: see text]. In this paper, we give effective bounds for [Formula: see text] and [Formula: see text].

Consider the switch chain on the set of $d$-regular bipartite graphs on $n$ vertices with $3\leq d\leq n^{c}$, for a small universal constant $c>0$. We prove that the chain satisfies a Poincaré inequality with a constant of order $O(nd)$; moreover, when $d$ is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $O_d(n\log n)$. We show that both results are optimal. The Poincaré inequality implies that in the regime $3\leq d\leq n^c$ the mixing time of the switch chain is at most $O\big((nd)^2 \log(nd)\big)$, improving on the previously known bound $O\big((nd)^{13} \log(nd)\big)$ due to Kannan, Tetali and Vempala and $O\big(n^7d^{18} \log(nd)\big)$ obtained by Dyer et al. The log-Sobolev inequality that we establish for constant $d$ implies a bound $O(n\log^2 n)$ on the mixing time of the chain which, up to the $\log n$ factor, captures a conjectured optimal bound. Our proof strategy relies on building, for any fixed function on the set of $d$-regular bipartite simple graphs, an appropriate extension to a function on the set of multigraphs given by the configuration model. We then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. While our method falls into a rich class of comparison techniques for Markov chains on different state spaces, the crucial feature of the method - dealing with chains with a large distortion between their stationary measures - is a novel addition to the theory., revision

We determine the asymptotic distribution of the p-rank of the sandpile groups of random bipartite graphs. We see that this depends on the ratio between the number of vertices on each side, with a threshold when the ratio between the sides is equal to 1/p. We follow the approach of Melanie Wood and consider random graphs as a special case of random matrices, and rely on a variant the definition of min-entropy given by Maples, in order to obtain useful results about these random matrices. Our results show that unlike the sandpile groups of Erdos-Renyi random graphs, the distribution of the sandpile groups of random bipartite graphs depends on the properties of the graph, rather than coming from some more general random group model.

Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with ⌊ 2 n / 3 ⌋ \lfloor 2n/3 \rfloor and ⌈ n / 3 ⌉ \lceil n/3 \rceil vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [ 0 , 1 ] \mathscr {L}^2[0,1] . The second conjecture claims that for any fixed m ≤ n 2 / 4 m \leq n^2/4 , if G G maximizes spread over all n n -vertex graphs with m m edges, then G G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms.

Bipartite graphs have been widely used to model the relationship between entities of different types, where vertices are partitioned into two disjoint sets/sides. Finding dense subgraphs in a bipartite graph is of great significance and encompasses many applications. However, none of the existing dense bipartite subgraph models consider similarity between vertices from the same side, and as a result, the identified results may include vertices that are not similar to each other. In this paper, we formulate the notion of similar-biclique which is a special kind of biclique where all vertices from a designated side are similar to each other, and aim to enumerate all similar-bicliques. The naive approach of first enumerating all maximal bicliques and then extracting all maximal similar-bicliques from them is inefficient, as enumerating maximal bicliques is time consuming. We propose a backtracking algorithm MSBE to directly enumerate maximal similar-bicliques, and power it by vertex reduction and optimization techniques. Furthermore, we design a novel index structure to speed up a time-critical operation of MSBE, as well as to speed up vertex reduction. Efficient index construction algorithms are also developed. Extensive experiments on 17 bipartite graphs as well as case studies are conducted to demonstrate the effectiveness and efficiency of our model and algorithms.

A bipartite graph G can be treated as a (1,1) bipartite graph in the sense that, no two vertices in the same part are at distance one from each other. A (2,2) bipartite graph is an extension of the above concept in which no two vertices in the same part are at distance two from each other. In this article, analogous to complete (1,1) bipartite graphs which have the maximum number of pairs of vertices having distance one between them, a complete (2,2) bipartite graph is defined as follows. A complete (2,2) bipartite graph is a graph which is (2,2) bipartite and has the maximum number of pairs of vertices (u,v) such that d(u,v)=2. Such graphs are characterized and their properties are studied. The expressions are derived for the determinant, the permanent and spectral properties of some classes of complete (2,2) bipartite graphs. A class of graphs among complete (2,2) bipartite graphs having golden ratio in their spectrum is obtained.

Maximal bicliques are effective to reveal meaningful information hidden in bipartite graphs. Maximal biclique enumeration (MBE) is challenging since the number of the maximal bicliques grows exponentially w.r.t. the number of vertices in a bipartite graph in the worst case. However, a large bipartite graph is usually very sparse, which is against the worst case and may lead to fast MBE algorithms. The uncharted opportunity is taking advantage of the sparsity to substantially improve the MBE efficiency for large sparse bipartite graphs. We observe that for a large sparse bipartite graph, a vertex u may converge to a few vertices in the same vertex set as u via its neighbours, which reveals that the enumeration scope for a vertex could be very small. Based on this observation, we propose novel concepts: unilateral coreness for individual vertices, unilateral order for each vertex set and unilateral convergence (ζ) for a large sparse bipartite graph, ζ could be a few thousand for a large sparse bipartite graph with hundreds of million edges. Using the unilateral order, every vertex with τ unilateral coreness only needs to check at most 2 τ combinations so that all maximal bicliques can be enumerated and τ is bounded by ζ, which leads to a novel MBE algorithm running in O * (2 ζ ). We then propose a batch-pivots technique to eliminate all enumerations resulting in non-maximal bicliques, which guarantees that every maximal biclique is reported in O (ζ e )-delay, where e is the number of edges. We devise novel data structures that allow storing subgraphs at omissible space for further speeding up MBE. Extensive experiments are conducted on synthetic and real large datasets to justify that our proposed algorithm is faster and more scalable than the existing algorithms.

As a fundamental data structure, graphs are ubiquitous in various applications. Among all types of graphs, signed bipartite graphs contain complex structures with positive and negative links as well as bipartite settings, on which conventional graph analysis algorithms are no longer applicable. Previous works mainly focus on unipartite signed graphs or unsigned bipartite graphs separately. Several models are proposed for applications on the signed bipartite graphs by utilizing the heuristic structural information. However, these methods have limited capability to fully capture the information hidden in such graphs. In this paper, we propose the first graph neural network on signed bipartite graphs, namely Polarity-based Graph Convolutional Network (PbGCN), for sign prediction task with the help of balance theory. We introduce the novel polarity attribute to signed bipartite graphs, based on which we construct one-mode projection graphs to allow the GNNs to aggregate information between the same type nodes. Extensive experiments on five datasets demonstrate the effectiveness of our proposed techniques.

Research on graph representation learning (a.k.a. embedding) has received great attention in recent years and shows effective results for various types of networks. Nevertheless, few initiatives have been focused on the particular case of embeddings for bipartite graphs. In this paper, we first define the graph embedding problem in the case of bipartite graphs. Next, we propose a taxonomy of approaches used to tackle this problem and draw a description of state-of-the-art methods. Then, we establish their pros and cons with respect to conventional network embeddings. Finally, we provide a description of available resources to lead experiments on the subject.

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent interest. We also prove a semicircle law for random $(d_1,d_2)$-biregular bipartite graphs when $\frac{d_1}{d_2}\to\infty$. As an application, we translate the results to adjacency matrices of uniformly distributed random regular hypergraphs., Comment: 45 pages, 5 figures, minor revision

By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs satisfying a weak expansion condition: when $d$ is constant, and the graph is a bipartite $\Omega( \log^2 d/d)$-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for $d>5$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a $d$-regular, bipartite $\alpha$-expander, with $\alpha>0$ fixed, we give an FPTAS for the hard-core model partition function at fugacity $\lambda=\Omega(\log d / d^{1/4})$. Finally we present an algorithm that applies to all $d$-regular, bipartite graphs, runs in time $\exp\left( O\left( n \cdot \frac{ \log^3 d }{d } \right) \right)$, and outputs a $(1 + o(1))$-approximation to the number of independent sets.

We say that a bipartite graph $G(A, B)$ with fixed parts $A$, $B$ is proximinal if there is a semimetric space $(X, d)$ such that $A$ and $B$ are disjoint proximinal subsets of $X$ and all edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A, B)$. It is proved that a bipartite graph $G$ is not isomorphic to any proximinal graph iff $G$ is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph $G$ is a disjoint union of complete bipartite graphs iff $G$ is isomorphic to a nonempty proximinal graph for an ultrametric space., 20 pages, 2 figures

Mining cohesive subgraphs on bipartite graphs is an important task. The k -bitruss is one of many popular cohesive subgraph models, which is the maximal subgraph where each edge is contained in at least k butterflies. The bitruss decomposition problem is to find all k -bitrusses for k ≥ 0. Dealing with large graphs is often beyond the capability of a single machine due to its limited memory and computational power, leading to a need for efficiently processing large graphs in a distributed environment. However, all current solutions are for a single machine and a centralized environment, where processors can access the graph or auxiliary indexes randomly and globally. It is difficult to directly deploy such algorithms on a shared-nothing model. In this paper, we propose distributed algorithms for bitruss decomposition. We first propose SC-HBD as the baseline, which uses H -function to define bitruss numbers and computes them iteratively to a fix point in parallel. We then introduce a subgraph-centric peeling method SC-PBD, which peels edges in batches over different butterfly complete subgraphs. We then introduce local indexes on each fragment, study the butterfly-aware edge partition problem including its hardness, and propose an effective partitioner. Finally we present the bitruss butterfly-complete subgraph concept, and divide and conquer DC-BD method with optimization strategies. Extensive experiments show the proposed methods solve graphs with 30 trillion butterflies in 2.5 hours, while existing parallel methods under shared-memory model fail to scale to such large graphs.

We design and implement an efficient parallel algorithm for finding a perfect matching in a weighted bipartite graph such that weights on the edges of the matching are large. This problem differs from the maximum weight matching problem, for which scalable approximation algorithms are known. It is primarily motivated by finding good pivots in scalable sparse direct solvers before factorization. Due to the lack of scalable alternatives, distributed solvers use sequential implementations of maximum weight perfect matching algorithms, such as those available in MC64. To overcome this limitation, we propose a fully parallel distributed memory algorithm that first generates a perfect matching and then iteratively improves the weight of the perfect matching by searching for weight-increasing cycles of length four in parallel. For most practical problems the weights of the perfect matchings generated by our algorithm are very close to the optimum. An efficient implementation of the algorithm scales up to 256 nodes (17,408 cores) on a Cray XC40 supercomputer and can solve instances that are too large to be handled by a single node using the sequential algorithm., Comment: 28 pages, 9 figures

Recently, graph-based methods have been widely applied to model fitting. However, in these methods, association information is invariably lost when data points and model hypotheses are mapped to the graph domain. In this paper, we propose a novel model fitting method based on co-clustering on bipartite graphs (CBG) to estimate multiple model instances in data contaminated with outliers and noise. Model fitting is reformulated as a bipartite graph partition behavior. Specifically, we use a bipartite graph reduction technique to eliminate some insignificant vertices (outliers and invalid model hypotheses), thereby improving the reliability of the constructed bipartite graph and reducing the computational complexity. We then use a co-clustering algorithm to learn a structured optimal bipartite graph with exact connected components for partitioning that can directly estimate the model instances (i.e., post-processing steps are not required). The proposed method fully utilizes the duality of data points and model hypotheses on bipartite graphs, leading to superior fitting performance. Exhaustive experiments show that the proposed CBG method performs favorably when compared with several state-of-the-art fitting methods.

Many authors investigated unmixed and bipartite graphs. There are many results obtained on these two types of graphs and show that a bipartite graph G satisfies Serre’s condition S2, then G is Cohen-Macaulay. This paper is a survey with some new results about very well-covered graphs, an unmixed bipartite graph.

A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources ("cakes") or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-floor(2n/3) maximin-share allocation of discrete bads among n agents., Comment: Appeared in Information Sciences, 587:164--187. But during the production, the main theorem text was deleted. The arXiv version is the correct one

A strong edge-coloring $\varphi$ of a graph $G$ assigns colors to edges of $G$ such that $\varphi(e_1)\ne \varphi(e_2)$ whenever $e_1$ and $e_2$ are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of the line graph of $G$. In 1990 Faudree, Schelp, Gy\'arf\'as, and Tuza conjectured that if $G$ is a bipartite graph with maximum degree 3 and sufficiently large girth, then $G$ has a strong edge-coloring with at most 5 colors. In 2021 this conjecture was disproved by Lu\v{z}ar, Ma\v{c}ajov\'{a}, \v{S}koviera, and Sot\'{a}k. Here we give an alternative construction to disprove the conjecture., Comment: 3.5 pages, 1 figure; second version extends the construction from the 3-regular case to the $k$-regular case, for each integer $k\ge 2$; third version incorporates referee feedback; to appear in Discrete Applied Math

The genus of the binomial random graph $G(n,p)$ is well understood for a wide range of $p=p(n)$. Recently, the study of the genus of the random bipartite graph $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour to that of the binomial random graph. In this paper we show that there is a threshold for planarity of the random bipartite graph at $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$., Comment: 20 pages, added remark 4.15 and fixed small errors

Negative samples in health and medical insurance domain refer to fraudulent or erroneous insurance claims that may include inconsistent diagnosis-procedure relations with respect to a medical coding system. Unfortunately, only a few datasets are publicly available for research in health insurance domain, yet none reports any negative claims. However, negative claims are essential not only to develop new machine learning approaches but also to test and validate automated artificial intelligence systems deployed by insurance providers. In this study, we introduce a synthetic negative claim generation procedure based on the bipartite graph representations of positive claims. Our empirical results demonstrate promising outcomes that will improve the development and evaluation processes of machine learning approaches in healthcare, where negative samples are required, but not available. Moreover, the proposed scheme can be applied to other domains, where bipartite graph representations are meaningful and negative samples are lacking.

In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph $G$ and an integer $k$ one has to decide whether there exists a perfect matching in $G$ with exactly $k$ red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly $k$ red edges, not a lot of work focuses on computing perfect matchings with almost $k$ red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS'23). It outputs a perfect matching with $k'$ red edges with the guarantee that $0.5k \leq k' \leq 1.5k$. In the present paper we aim at approximating the number of red edges without exceeding the limit of $k$ red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with $k'$ red edges such that $k/3 \leq k' \leq k$.

We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively.

Bipartite graphs model relationships between two different sets of entities, like actor-movie, user-item, and author-paper. The butterfly, a 4-vertices 4-edges $2\times 2$ bi-clique, is the simplest cohesive motif in a bipartite graph and is the fundamental component of higher-order substructures. Counting and enumerating the butterflies offer significant benefits across various applications, including fraud detection, graph embedding, and community search. While the corresponding motif, the triangle, in the unipartite graphs has been widely studied in both static and temporal settings, the extension of butterfly to temporal bipartite graphs remains unexplored. In this paper, we investigate the temporal butterfly counting and enumeration problem: count and enumerate the butterflies whose edges establish following a certain order within a given duration. Towards efficient computation, we devise a non-trivial baseline rooted in the state-of-the-art butterfly counting algorithm on static graphs, further, explore the intrinsic property of the temporal butterfly, and optimize the process with a compact data structure and smart pruning strategies. The time complexity is proved to be significantly reduced without compromising on space efficiency. In addition, we generalize our algorithms to practical streaming settings and multi-core computing architectures. Our extensive experiments on 11 large-scale real-world datasets demonstrate the efficiency and scalability of our solutions.

Community search over bipartite graphs has attracted significant interest recently. In many applications such as the user–item bipartite graph in e-commerce and customer–movie bipartite graph in movie rating website, nodes tend to have attributes. However, the previous community search algorithms on bipartite graphs ignore attributes, thus making them to return results with poor cohesion with respect to their node attributes. In this paper, we study the community search problem on attributed bipartite graphs. Given a query vertex [Formula: see text], we aim to find the attributed [Formula: see text]-communities of [Formula: see text], where the structure cohesiveness of the community is described by the [Formula: see text]-core model, and the attribute similarity of two groups of nodes in the subgraph is maximized. In order to retrieve attributed communities from bipartite graphs, we first propose a basic algorithm composed of two steps: the generation and verification of candidate keyword sets, and then two improved query algorithms Inc and Dec are proposed. Inc is proposed considering the anti-monotonicity property of attributed bipartite graphs, then we adopt different generating methods and verify the order of candidate keyword sets and propose the Dec algorithm. After evaluating our solutions on eight large graphs, the experimental results demonstrate that our methods are effective and efficient in querying the attributed communities on bipartite graphs.

We present a ( 1 + k k + 2 ) -approximation algorithm for the Maximum k - dependent Set problem on bipartite graphs for any k ≥ 1 . For a graph with n vertices and m edges, the algorithm runs in O ( k m n ) time and improves upon the previously best-known approximation ratio of 1 + k k + 1 established by Kumar et al. (2014). Our proof also indicates that the algorithm retains its approximation ratio when applied to the (more general) class of Konig–Egervary graphs.