Bipartite Graph Matching Research Articles

Adversarial resilience of matchings in bipartite random graphs

Adversarial resilience of matchings in bipartite random graphs

A Semi-Matching Based Load Balancing Scheme for Dense IEEE 802.11 WLANs

A load balancing mechanism can adjust the load distribution among access points (APs) and improve resource utilization for dense wireless local area networks (WLANs). In this paper, we propose a semi-matching-based load balancing scheme for the IEEE 802.11 dense WLANs. The proposed scheme runs in a centralized controller. The controller judges whether the load is unevenly distributed according to the collected channel busy time ratio information of the entire network, and triggers the load balancing mechanism accordingly. In order to realize load balancing among APs and maximize the overall network throughput, we model the station to AP association problem as a weighted bipartite graph matching problem and find the optimal semi-matching using the Kuhn–Munkres (K-M) algorithm. Simulation results show that the proposed scheme achieves performance improvement comparing with traditional schemes.

Subset matching and edge coloring in bipartite graphs

The focus of this paper is on finding a matching in a bipartite graph such that a given subset of vertices are matched. This is called subset matching and generalizes perfect matchings. We prove a necessary and sufficient condition for the existence of a subset matching in bipartite graphs. The proof is algorithmic and based on combination of two matchings. Remarkably, the necessary and sufficient condition always holds when the subset is composed of the vertices with maximum degree. This in turn leads to a simple algorithm that finds an optimal edge coloring in bipartite graphs with no need to transform the bipartite graph into a regular one.

On rainbow matchings in bipartite graphs

We present recent results regarding rainbow matchings in bipartite graphs. Using topological methods we address a known conjecture of Stein and show that if Kn,n is partitioned into n sets of size n, then a partial rainbow matching of size 2n/3 exists. We generalize a result of Cameron and Wanless and show that for any n matchings of size n in a bipartite graph with 2n vertices there exists a full matching intersecting each matching at most twice. We show that any n matchings of size approximately 3n/2 have a rainbow matching of size n. Finally, we show the uniqueness of the extreme case for a theorem of Drisko and provide a generalization of Drisko's theorem.

An elementary integrality proof of Rothblum’s stable matching formulation

In this paper we provide a short new proof for the integrality of Rothblum’s linear description of the convex hull of incidence vectors of stable matchings in bipartite graphs. The key feature of our proof is to show that extreme points of the formulation must have a 0,1-component.

Perfect Matchings of Regular Bipartite Graphs

AbstractLet G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.

WSDL Retrieval for Web Services Based on Hybrid SLVM

Recently, two operable WSDL retrieval approaches, bipartite-graph matching and KbSM, were developed for Web service discovery. But their models and similarity metrics of WSDL ignore some term or semantic feature, and involve formal method problem of representation or difficulty of parameter verification. SLVM approaches depend on statistical term measures to implement XML document representation. Consequently, they ignore the lexical semantics and the relevant mutual information, leading to dataset analysis errors. This work proposed a service retrieval method, hybrid SLVM of WSDL, to address above problems in feature extraction. This method constructed a lexical semantic spectrum using WordNet for characterizing the lexical semantics, and built a special term spectrum based on TF-IDF. Then, feature matrix for WSDL representation was built on the hybrid SLVM. Applying to NWKNN algorithm, on dataset OWLS-TC-v2, our method achieves better F1 measure and query precisions than bipartite-graph matching and KbSM.

Anti-Ramsey numbers for matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(Kn, H) was introduced by Erdős, Simonovits and Sós in 1973, which is defined to be the maximum number of colors in an edge coloring of the complete graph Kn without any rainbow H. Later, the anti-Ramsey numbers for several special graph classes in complete are determined. Moreover, researchers generalized the host graph Kn to other graphs, in particular, to complete bipartite graphs and regular bipartite graphs. Li and Xu (2009) [18] proved that: Let G be a k-regular bipartite graph with n vertices in each partite set, then AR(G,mK2)=k(m−2)+1 for all m ≥ 2, k ≥ 3 and n>3(m−1). In this paper, we consider the anti-Ramsey number for matchings in 3-regular bipartite graphs. By using the known result that the vertex cover equals the size of maximum matching in bipartite graphs, we prove that AR(G,mK2)=3(m−2)+1 for n>32(m−1) when G is a 3-regular bipartite graph with n vertices in each partite set.

On Computing an Optimal Semi-matching

The problem of finding an optimal semi-matching is a generalization of the problem of finding classical matching in bipartite graphs. A semi-matching in a bipartite graph G = (U, V, E) with n vertices and m edges is a set of edges M ⊆ E, such that each vertex in U is incident to at most one edge in M. An optimal semi-matching is a semi-matching with degM(u) = 1 for all u ∈ U and the minimal value of \(\sum_{v \in V} \frac{deg_M(v).(deg_M(v)+1)}2\). We propose a schema that allows a reduction of the studied problem to a variant of the maximum bounded-degree semi-matching problem. The proposed schema yields to two algorithms for computing an optimal semi-matching. The first one runs in time \(O(\sqrt{n} \cdot m \cdot \log{n})\) that is the same as the time complexity of the currently best known algorithm. However, our algorithm uses a different approach that enables some improvements in practice (e.g. parallelization, faster algorithms for special graph classes). The second one is randomized and it computes an optimal semi-matching with high probability in O(n ω ·log1 + o(1) n), where ω is the exponent of the best known matrix multiplication algorithm. Since ω ≤ 2.38, this algorithms breaks through O(n 2.5) barrier for dense graphs.

Improved quadratic time approximation of graph edit distance by combining Hausdorff matching and greedy assignment

Approximation of graph edit distance in polynomial time enables us to compare large, arbitrarily labeled graphs for structural pattern recognition. In a recent approximation framework, bipartite graph matching (BP) has been proposed to reduce the problem of edit distance to a cubic-time linear sum assignment problem (LSAP) between local substructures. Following the same line of research, first attempts towards quadratic-time approximation have been made recently, including a lower bound based on Hausdorff matching (Hausdorff Edit Distance) and an upper bound based on greedy assignment (Greedy Edit Distance). In this paper, we compare the two approaches and derive a novel upper bound (BP2) which combines advantages of both. In an experimental evaluation on the IAM graph database repository, we demonstrate that the proposed quadratic-time methods perform equally well or, quite surprisingly, in some cases even better than the cubic-time method.

A non-rigid 3D model retrieval method based on scale-invariant heat kernel signature features

The number of non-rigid 3D models increases steadily in various areas. It is imperative to develop efficient retrieval system for 3D non-rigid models. As we know, global features fail to consistently describe the intra-class variability of non-rigid 3D models, the local features are more effective than global features for the retrieval of non-rigid 3D models. In this paper, we use Heat Kernel Signature (HKS) as the local features to represent non-rigid 3D models and further propose the retrieval method based on scale-invariant local features. Firstly, we extract key-points at multiple scales automatically. Then, the HKS local features are computed for each key-point. However, the HKS features are sensitive to scale. In order to solve this problem, we convert the scale problem into the translation problem using the diffusion Wavelets transform. To solve the translation problem, we use a kind of histogram equalization technique. Finally, we use the bipartite graph matching algorithm to compute similarity between the 3D models. Experimental results on two public benchmarks show that our method outperforms state-of-the-art methods for non-rigid 3D models retrieval.

Fast Algorithm to Find 2-Factor of Minimum Weight

The paper considers the minimization of the sum of weights of edges forming a subset U? ? U of the set of disjoint simple cycles at vertices ? ∈ V in graph H = (V,U) and covering V. The problem under study (2-f problem) is polynomially solvable in algorithms that are characterized by technical difficulties that hinder the acceleration of computing. The 2-f problem is solved by reducing it to a simpler bipartite case. The result is represented by a perfect matching of bipartite graph corresponding to the solution of the assignment problem, where each circuit of cyclic expansion contains at least three arcs.

3D object retrieval with multi-feature collaboration and bipartite graph matching

In this paper, we propose a novel 3D object retrieval with features collaboration and bipartite graph matching strategies. We explored the essential characters of 3D object in a view-based retrieval framework, which extracts complement descriptors from both the contour and the interior region of 3D object effectively. Specifically, a greedy bipartite graph matching algorithm is employed. With the bipartite graph matching and feature concatenation, significant performance improvement is achieved in the 3D object retrieval task. The proposed method is evaluated by the third party on the data set comprising more than 500 3D objects and achieves the best performance for SHREC’15 challenge.

A fast algorithm to construct a representation for transversal matroids

Transversal matroids were introduced in the mid 1960s and unified many results in transversal theory. Piff and Welsh proved that a transversal matroid is representable over all sufficiently large fields. To date, their merge algorithm is the only known algorithm to construct a representation matrix for a given transversal matroid. In this paper, a new algorithm to construct a representation matrix for a given transversal matroid is proposed that is faster than the Piff–Welsh algorithm. Let $$G=(V^{(1)}\dot{\cup }V^{(2)},E)$$ be the bipartite graph representing a transversal matroid and $${\mathcal {M}}$$ the set of all complete matchings of G. The time complexity of the proposed algorithm is $$O\left( |V^{(1)}|^{1/2}|E| + |V^{(1)}| |{\mathcal {M}}|\right) $$ . This algorithm makes use of complete matchings of bipartite graphs instead of matrix determinants, and an enumeration algorithm is used to find these matchings.

Predicting the correctness of node assignments in bipartite graph matching

Due to their high representational power and flexibility, graphs have emerged as a widely used data structure for complex pattern representation. The advantages of graphs are, however, at the expense of a substantial increase of the computational complexity of the associated algorithms. For instance, exact graph similarity or distance can be computed in exponential time only. An algorithmic framework based on bipartite graph matching (allowing graph dissimilarity computation in cubic time) has been presented recently. However, this fast computation leads in general to some incorrect node assignments with respect to a globally optimal graph matching. In this paper we present a novel methodology for predicting which of the node assignments are actually incorrectly assigned by this approximation. To this end, a comprehensive set of features – which numerically characterize the assignments – is defined and extracted from the underlying graphs. These features are in turn used for training a statistical classifier, which can eventually be employed for predicting unseen assignments. The power and applicability of our novel approach is empirically verified on eight real world data sets.

Principled Graph Matching Algorithms for Integrating Multiple Data Sources

This paper explores combinatorial optimization for problems of max-weight graph matching on multi-partite graphs, which arise in integrating multiple data sources. In the most common two-source case, it is often desirable for the final matching to be one-to-one; the database and statistical record linkage communities accomplish this by weighted bipartite graph matching on similarity scores. Such matchings are intuitively appealing: they leverage a natural global property of many real-world entity stores—that of being nearly deduped—and are known to provide significant improvements to precision and recall. Unfortunately, unlike the bipartite case, exact max-weight matching on multi-partite graphs is known to be NP-hard. Our two-fold algorithmic contributions approximate multi-partite max-weight matching: our first algorithm borrows optimization techniques common to Bayesian probabilistic inference; our second is a greedy approximation algorithm. In addition to a theoretical guarantee on the latter, we present comparisons on a real-world entity resolution problem from Bing significantly larger than typically found in the literature, on publication data, and on a series of synthetic problems. Our results quantify significant improvements due to exploiting multiple sources, which are made possible by global one-to-one constraints linking otherwise independent matching sub-problems. We also discover that our algorithms are complementary: one being much more robust under noise, and the other being simple to implement and very fast to run.

Stochastic Enumeration Method for Counting Trees

The problem of estimating the size of a backtrack tree is an important but hard problem in the computational sciences. An efficient solution of this problem can have a major impact on the hierarchy of complexity classes. The first randomized procedure, which repeatedly generates random paths through the tree, was introduced by Knuth. Unfortunately, as was noted by Knuth and a few other researchers, the estimator can introduce a large variance and become ineffective in the sense that it underestimates the cost of the tree. Recently, a new sequential algorithm called Stochastic Enumeration (SE) method was proposed by Rubinstein et al. The authors showed numerically that this simple algorithm can be very efficient for handling different counting problems, such as counting the number of satisfiability assignments and enumerating the number of perfect matchings in bipartite graphs. In this paper we introduce a rigorous analysis of SE and show that it results in significant variance reduction as compared to Knuth’s estimator. Moreover, we establish that for almost all random trees the SE algorithm is a fully polynomial time randomized approximation scheme (FPRAS) for the estimation of the overall tree size.

Graph Edit Distance: Moving from global to local structure to solve the graph-matching problem

Classical graph approaches for pattern recognition applications rely on computing distances between graphs in a certain graph domain. That is, the distance between two graphs is obtained by directly optimising some objective function, which consider node and edge attributes. Bipartite Graph Matching was first published in a journal in 2009 and new versions have appeared to speed up its runtime such as the Fast Bipartite Graph Matching. This algorithm is based on defining a cost matrix between the whole nodes of both graphs and solving the node correspondence through a linear assignment method. To construct the matrix, several local structures can be defined from the simplest one (only the node) to the most complex (a whole clique or eigenvector structure). In this paper, we propose eight different options and we show that the type of local structure and the distance defined between these structures is relevant for the runtime and classification ratio.

Multi-View Video Summarization Using Bipartite Matching Constrained Optimum-Path Forest Clustering

The task of multi-view video summarization is to efficiently represent the most significant information from a set of videos captured for a certain period of time by multiple cameras. The problem is highly challenging because of the huge size of the data, presence of many unimportant frames with low activity, inter-view dependencies, and significant variations in illumination. In this paper, we propose a graph-theoretic solution to the above problems. Semantic feature in form of visual bag of words and visual features like color, texture, and shape are used to model shot representative frames after temporal segmentation . Gaussian entropy is then applied to filter out frames with low activity. Inter-view dependencies are captured via bipartite graph matching. Finally, the optimum-path forest algorithm is applied for the clustering purpose. Subjective as well as objective evaluations clearly indicate the effectiveness of the proposed approach.

Improving bipartite graph matching by assessing the assignment confidence

Due to the ability of graphs to represent properties of entities and binary relations at the same time, a growing interest in this representation formalism can be observed in various fields of pattern recognition. The availability of a distance measure is a basic requirement for pattern recognition. For graphs, graph edit distance is still one of the most popular distance measures. In the present paper we substantially improve the distance accuracy of a recent framework for the approximation of graph edit distance. The basic idea of our novel approach is to manipulate the initial assignment returned by the approximation algorithm such that the individual assignments are ordered according to their individual confidence. Next, the individual assignments are post processed in this specific order. In an experimental evaluation we show that the order of the assignments plays a crucial role for the resulting distance accuracy. Moreover, we empirically verify that our novel generalization is able to generate approximations which are very near to the exact edit distance (in contrast with the original framework).