Bipartite Graph Matching Research Articles

Research on Matching and Vertex Cover Problems in Bipartite Graphs using Simplex Method

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.

Efficient joint object matching via linear programming

Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer linear programming (ILP) problem. To construct strong LP relaxations, we study the facial structure of the convex hull of the feasible region of this ILP, which we refer to as the joint matching polytope. We present an exponential family of facet-defining inequalities that can be separated in strongly polynomial time, hence obtaining a partial characterization of the joint matching polytope that is both tight and cheap to compute. To analyze the theoretical performance of the proposed LP relaxations, we focus on permutation group synchronization, an important special case of joint object matching. We show that under the random corruption model for the input maps, a simple LP relaxation, that is, an LP containing only a very small fraction of the proposed facet-defining inequalities, recovers the ground truth with high probability if the corruption level is below 40%. Finally, via a preliminary computational study on synthetic data, we show that the proposed LP relaxations outperform a popular SDP relaxation both in terms of recovery and tightness.

Graph Neural Networks for Cross-Camera Data Association

Cross-camera image data association is essential for many multi-camera computer vision tasks, such as multi-camera pedestrian detection, multi-camera multi-target tracking, 3D pose estimation, etc. This association task is typically modeled as a bipartite graph matching problem and often solved by applying minimum-cost flow techniques, which may be computationally demanding for large data. Furthermore, cameras are usually treated by pairs, obtaining local solutions, rather than finding a global solution at once for all multiple cameras. Other key issue is that of the affinity function: the widespread usage of non-learnable pre-defined distances, such as the Euclidean and Cosine ones. This paper proposes an effective approach for cross-camera data-association focused on a global solution, instead of processing cameras by pairs. To avoid the usage of fixed distances and thresholds, we leverage the connectivity of Graph Neural Networks, previously unused in this scope, using a Message Passing Network to jointly learn features and similarity functions. We validate the proposal for pedestrian cross-camera association, showing results over the EPFL multi-camera pedestrian dataset. Our approach considerably outperforms the literature data association techniques, without requiring to be trained in the same scenario in which it is tested. Our code is available at <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><uri>https://www-vpu.eps.uam.es/publications/gnn</uri></monospace>

PPO-TA: Adaptive task allocation via Proximal Policy Optimization for spatio-temporal crowdsourcing

With the pervasiveness of dynamic task allocation in sharing economy applications, the online bipartite graph matching has attracted people’s increasing attention to its research in recent years. Among its application in sharing economy, the crowdsourcing allocate the tasks to workers dynamically. There are still three main problems that need to be addressed from previous studies. (1) These task allocation algorithms usually ignore the long-term utility on crowdsourcing platforms. (2)The current research works show that it has low allocation numbers. (3) Due to the low number of allocations, it becomes difficult to improve total allocation utilities. In this paper, we considered the long-term utility and drawed an idea of dynamic delay bipartite graph matching(DDBM). We proposed a Policy Gradient Based Discrete Threshold Task Allocation algorithm (DTTA) and a Proximal Policy Optimization Based Continuous Threshold Task Allocation algorithm (PPOTA) to solve these problems. The extensive experimental results on two real datasets demonstrate that the proposed algorithms are superior to the existing algorithms in both effectiveness and efficiency.

Long-Term Matching Optimization With Federated Neural Temporal Difference Learning in Mobility-on-Demand Systems

Efficient bipartite graph matching of orders and drivers is a central operational problem in industrial mobility-on-demand (MOD) systems. Traditional studies adopt pure combinatorial optimization models for order-and-driver matching, which do not consider the long-term rewards of the dynamic MOD decision making process. Toward long-term optimization, this article presents a systemic paradigm of online matching with federated neural temporal difference learning (FedTDLearning), which encompasses learning and matching phases. During the learning phase, the long-term matching process is modeled as a Markov decision process, which is typically solved by employing data-driven reinforcement learning in an offline central training scheme. Massive amounts of data would be generated on the network by industrial MOD systems. It is impossible to send all the large-scale industrial data to the cloud server for centralized model training due to network bandwidth limitations and safety concerns. Therefore, a generic and innovative FedTDLearning is proposed to achieve long-term matching in a distributed manner. During the matching phase, a real-time bipartite matching optimization problem is formulated to maximize the learned spatiotemporal value and minimize the pickup distance, which is reducible to the minimum-cost maximum weight bipartite graph matching problem. A distance-learned-value ratio algorithm is proposed to find an optimal matching in the bipartite graph based on the joint optimization of FedTDLearning and the combinatorial fractional programming approach. Furthermore, to pursue optimal computation efficiency, we solve the real-time matching problem by constructing a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -nearest neighbor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(k$ </tex-math></inline-formula> NN) bipartite graph where each order is connected with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> NN drivers. Using openly available real-world data, the prototype system and experimental evaluations show that our proposed algorithms have effective problem-solving capability in practice.

Test Optimization in Memristor Crossbars Based on Path Selection

Memristors have recently shown significant promise in designing memory and logic subsystems. A 2D-crossbar architecture built with memristor arrays provides a convenient platform for storing multivalued memory states by utilizing the analog variation of current-induced resistance through these cells. The integration of CMOS components with non-CMOS memristor cells further enhances the scope of their applications to various complex system designs. However, present-day memristors by virtue of their inherent structure are prone to various manufacturing defects and sensitive to operational modalities. Existing techniques for testing memristor arrays are either ad hoc in nature or suited for application-specific designs with little concern for optimizing test time. In this work, we envisage a 2-D memristor crossbar as a network and identify certain paths that are suitable for fault sensitization. For full-size square and rectangular memristive crossbars, the proposed method optimizes test time using a path-based technique guided by maximum matching in bipartite graphs. An integer linear programming (ILP) formulation is then used to solve the problem for a general crossbar, either full or incomplete. Simulation results with LTspice demonstrate the effectiveness and superiority of the method to the prior art in terms of test time and fault coverage.

Secrecy rate maximization for 6G cognitive satellite-UAV networks

To cover remote areas where terrestrial cellular networks may not be available, non-terrestrial infrastructures such as satellites and unmanned aerial vehicles (UAVs) can be utilized in the upcoming sixth-generation (6G) era. Considering the spectrum scarcity problem, satellites and UAVs need to share the spectrum to save costs, leading to a cognitive satellite-UAV network. Due to the openness of both satellite links and UAV links, communication security has become a major concern in cognitive satellite-UAV networks. In this paper, we safeguard a cognitive satellite-UAV network from a physical layer security (PLS) perspective. Using only the slowly-varying large-scale channel state information (CSI), we jointly allocate the transmission power and subchannels to maximize the secrecy sum rate of UAV users. The optimization problem is a mixed integer nonlinear programming (MINLP) problem with coupling constraints. We propose a heuristic algorithm which relaxes the coupling constraints by the penalty method and obtains a sub-optimal low-complexity solution by utilizing random matrix theory, the max-min optimization tool, and the bipartite graph matching algorithm. The simulation results corroborate the superiority of our proposed scheme.

Energy-Efficient Resource Allocation for Short Packet Transmission in MISO Multicarrier NOMA

In this article, we investigate energy efficiency (EE) maximization of short packet transmission (SPT) in a downlink multiple-input single-output (MISO) multi-carrier non-orthogonal multiple access (MC-NOMA) communication system. We use the ratio of the effective throughput to the power as a performance metric to formulate a non-convex mixed-integer nonlinear programming (MINLP) problem. Then, a low-complexity three-step optimization framework is proposed for MINLP optimization. Under a given maximum transmit power and channel blocklength of each subcarrier, a block coordinate descent (BCD) based method is proposed to jointly optimize the power allocation coefficient, transmission rate, and transmit power within a single subcarrier. Then, we transform the subcarrier allocation problem into a weighted bipartite graph matching problem and obtain the optimal subcarrier allocation scheme using the Kuhn-Munkres (KM) algorithm. Finally, a dynamic programming (DP) method is proposed to obtain the optimal channel blocklength allocation scheme. The simulation results validate the effectiveness of the proposed three-step optimization framework and indicate that the MC-NOMA system achieves higher EE and higher effective throughput than the orthogonal frequency-division multiple access (OFDMA) system in SPT.

Link-State Aware Hybrid Routing in the Terrestrial–Satellite Integrated Network

In this paper, we study data transmission in the Terrestrial-Satellite Integrated Network (TSIN), where terrestrial networks and satellites are combined together to provide seamless global network services for ground users. However, efficiency of the data transmission is limited by the time-varying inter-satellite link connection and intermittent terrestrial-satellite link connection. Therefore, we propose a link-state aware hybrid routing algorithm, which selects the integrated data transmission path adaptively in this paper. First of all, a space-time topology model is constructed to represent the dynamic link connections in TSIN. Thus, the transmission delay can be analyzed accordingly, and the data transmission problem can then be formulated. To balance the effectiveness and accuracy of searching a hybrid path, we carefully discuss the optimization of space-time topology updating, and propose an inter-satellite link selection algorithm. For the terrestrial-satellite link in hybrid routing, the data transmission problem is transformed into a weighted bipartite graph matching problem and solved with a Kuhn-Munkres-based link selection algorithm. To verify the effectiveness of our proposed routing algorithm, extensive simulations are conducted based on a realistic Hongyun constellation project. Results show that the network performance is improved with respect to data transmission delay, packet loss rate, and throughput.

Recognition of Targets in SAR Images Based on a WVV Feature Using a Subset of Scattering Centers.

This paper proposes a robust method for feature-based matching with potential for application to synthetic aperture radar (SAR) automatic target recognition (ATR). The scarcity of measured SAR data available for training classification algorithms leads to the replacement of such data with synthetic data. As attributed scattering centers (ASCs) extracted from the SAR image reflect the electromagnetic phenomenon of the SAR target, this is effective for classifying targets when purely synthetic SAR images are used as the template. In the classification stage, following preparation of the extracted template ASC dataset, some of the template ASCs were subsampled by the amplitude and the neighbor matching algorithm to focus on the related points of the test ASCs. Then, the subset of ASCs were reconstructed to the world view vector feature set, considering the point similarity and structure similarity simultaneously. Finally, the matching scores between the two sets were calculated using weighted bipartite graph matching and then combined with several weights for overall similarity. Experiments on synthetic and measured paired labeled experiment datasets, which are publicly available, were conducted to verify the effectiveness and robustness of the proposed method. The proposed method can be used in practical SAR ATR systems trained using simulated images.

Automated Vehicle to Vehicle Conflict Analysis at Signalized Intersections by Camera and LiDAR Sensor Fusion

This research presents a robust approach for automated safety diagnosis using sensor fusion techniques. This work fuses the outputs of a roadside low-resolution camera and a solid-state LiDAR. For vehicle classification and detection in videos, the YOLO-v5 object detection model was utilized. The raw 3D point clouds generated by the LiDAR are processed by two manual steps—ground plane transformation and background segmentation, and two real-time steps—foreground clustering and bounding box fitting. Taking the generated 2D bounding boxes of both camera and LiDAR, we associate the common bounding box pairs by thresholding on the Euclidean distance threshold of 6 ft between the centroid pairs. We perform weighted measurement updates based on the root mean square error of each of the sensor’s detection compared with manually labeled ground truths. The fused measurements are tracked by using linear constant velocity Kalman filter. With the generated trajectories, we compute post encroachment time at pixel-level conflicts based on the generated vehicle trajectories. We have proposed a complete bipartite graph-matching strategy of vehicle parts along with the conflict angle to obtain conflict types—rear-end, sideswipe, head-on, and angle conflict. A case study on a signalized intersection is presented. The output of the proposed framework performs with 97.384% precision and 95.316% recall. It is better than both single-sensor-based systems in relation to detection count and localization. It is expected that the proposed method can be employed to diagnose road safety problems and inform the required countermeasures.

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb Counting Bases, Ph.D. thesis, University of Waterloo, 2004 introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy--Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities. Our study then turns to algorithmic aspects. We observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey SIAM J. Comput., 39 (2009), pp. 679--702 and Cheung, Lau, and Leung ACM Trans. Algorithms, 10 (2014), pp. 1--26 can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities. Our algorithms make use of Frank's weight-splitting lemma for the weighted matroid intersection problem and the algebraic optimality criterion of the weighted linear matroid parity problem given by Iwata and Kobayashi SIAM J. Comput., 51 (2002) pp. STOC17-238--STOC17-280.

Perfect matchings in inhomogeneous random bipartite graphs in random environment

In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erdős-Rényi random bipartite graphs in a random environment. We show that the expected number of perfect matchings obeys a precise asymptotic.

Vehicle Re-identification for Lane-level Travel Time Estimations on Congested Urban Road Networks Using Video Images

The provision of lane-level travel time information can enable accurate traffic control and route guidance in urban roads with distinctive traffic conditions among lanes. However, few studies in the literature have been conducted to estimate lane-level travel time distributions. This study proposes a new vehicle re-identification (V-ReID) method for estimating lane-level travel time distributions using video images from widely deployed surveillance cameras. In the proposed method, a lane-based bipartite graph matching is introduced to obtain optimal matches between upstream and downstream vehicles by considering lane-level traffic conditions and vehicles’ lane changing behaviors and visual features. A lane-based travel time estimation technique is introduced to real-time estimate full spectrum of lane-level distribution parameters, including not only the mean but also the standard deviation and the distribution type. A comprehensive case study is carried out on a congested urban road in Hong Kong. Results of case study show that the proposed method outperforms the state-of-the-art link-based V-ReID method and is capable for providing accurate lane-level travel time distribution information on congested urban roads.

Prophet Inequality for Bipartite Matching: Merits of Being Simple and Nonadaptive

We consider the Bayesian online selection problem of a matching in bipartite graphs, that is, the weighted online matching problem where the edges arrive online and edge weights are generated from a known distribution. This setting corresponds to the intersection of two matroids in the work of Kleinberg and Weinberg [ 40 ] and Feldman et al. [ 27 ]. We study a simple class of nonadaptive policies that we call vertex-additive policies. A vertex-additive policy assigns static prices to every vertex in the graph and accepts only those edges whose weight exceeds the sum of the prices on the edge endpoints. We show that there exists a vertex-additive policy with the expected payoff of at least one-third of the prophet’s payoff and present a gradient descent algorithm that quickly converges to the desired vector of vertex prices. Our results improve on the adaptive online policies of Kleinberg and Weinberg and Feldman et al. for the intersection of two matroids in two ways: our policy is nonadaptive and has a better approximation guarantee of 3 instead of the previous guarantees of 5.82 in Kleinberg and Weinberg and 5.43 in Feldman et al. We give a complementary lower bound of 2.25 for any online algorithm in the bipartite matching setting. Funding: This work was supported by Fundamental Research Funds of the Central Universities in China, a Science and Technology Innovation 2030 major project [“New Generation of Artificial Intelligence,” Project 2018AAA0100903], the Shanghai Municipal Education Commission Innovation Program, and the Program for Innovative Research Team of the Shanghai University of Finance and Economics (IRTSHUFE).

The n-queens completion problem

An n-queens configuration is a placement of n mutually non-attacking queens on an ntimes n chessboard. The n-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be completed to an n-queens configuration. In this paper, we study an extremal aspect of this question, namely: how small must a partial configuration be so that a completion is always possible? We show that any placement of at most n/60 mutually non-attacking queens can be completed. We also provide partial configurations of roughly n/4 queens that cannot be completed and formulate a number of interesting problems. Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.

Rainbow matchings in an edge-colored planar bipartite graph

In this paper, we consider the existence of rainbow matchings in maximal bipartite planar graphs. We determine the maximum number of colors appearing in an edge-coloring of maximal bipartite planar graphs with a Hamilton cycle which does not contain any rainbow k-matching. The bounds given are best possible.

Phenotype Tracking of Leafy Greens Based on Weakly Supervised Instance Segmentation and Data Association

Phenotype analysis of leafy green vegetables in planting environment is the key technology of precision agriculture. In this paper, deep convolutional neural network is employed to conduct instance segmentation of leafy greens by weakly supervised learning based on box-level annotations and Excess Green (ExG) color similarity. Then, weeds are filtered based on area threshold, K-means clustering and time context constraint. Thirdly, leafy greens tracking is achieved by bipartite graph matching based on mask IoU measure. Under the framework of phenotype tracking, some time-context-dependent phenotype analysis tasks such as growth monitoring can be performed. Experiments show that the proposed method can achieve 0.95 F1-score and 76.3 sMOTSA (soft multi-object tracking and segmentation accuracy) by using weakly supervised annotation data. Compared with the fully supervised approach, the proposed method can effectively reduce the requirements for agricultural data annotation, which has more potential in practical applications.

Technical Perspective

The optimal assignment problem is a classic combinatorial optimization problem. Given a set of n agents A, a set T of m tasks, and an n×m cost matrix C, the objective is to find the matching between A and T, which minimizes or maximizes an aggregate cost of the assigned agent-task pairs. In its standard definition, n = m and we are looking for the 1-to-1 matching with the minimum total cost. From a graph theory perspective, this is a weighted bipartite graph matching problem. A classic algorithm for solving the assignment problem is the Hungarian algorithm (a.k.a. Kuhn-Munkres algorithm) [3], which bears a O(n3) computational cost (assuming that n = m); this is the best run-time of any strongly polynomial algorithm for this problem. There are many variants of the assignment problem, which differ in the optimization objective (i.e., minimize/maximize an aggregate cost, achieve a stable matching, maximize the number of agents matched which their top preferences, etc.) and in whether there are constraints on the number of matches for each agent or task.

Coverage Enhancement Algorithm for WSNs Based on Vampire Bat and Improved Virtual Force

Coverage enhancement is an important strategy to avoid coverage holes caused by the random deployment of sensor nodes. Given the harsh physical environments of wireless sensor networks (WSNs), which impede the energy supplement and recovery of sensor nodes, the motivation of our research is to repair the coverage holes and reduce the energy consumption during the deployment of sensor nodes. Firstly, by using cellular grids to stack the monitoring area, a bipartite graph model between the nodes and the cellular grids is constructed to minimize and balance the moving distance during the redeployment. Secondly, the vampire bat algorithm is introduced for bipartite graph matching. Finally, the coverage effect and moving distance are further optimized based on the improved virtual force. The simulation results show that the coverage rate is increased by 5.43&#x0025;, 3.25&#x0025;, and 1.63&#x0025;, and the moving distance is reduced by 33.64&#x0025;, 3.66&#x0025;, and 2.01&#x0025; when compared with COSH, VF-IALO, and HA, respectively. In addition, the proposed algorithm shows good performance in terms of uniformity of moving distance and maximum moving distance.