Bipartite Matching Problem Research Articles

A neural algorithm for computing bipartite matchings

Finding optimal bipartite matchings-e.g., matching medical students to hospitals for residency, items to buyers in an auction, or papers to reviewers for peer review-is a fundamental combinatorial optimization problem. We found a distributed algorithm for computing matchings by studying the development of the neuromuscular circuit. The neuromuscular circuit can be viewed as a bipartite graph formed between motor neurons and muscle fibers. In newborn animals, neurons and fibers are densely connected, but after development, each fiber is typically matched (i.e., connected) to exactly one neuron. We cast this synaptic pruning process as a distributed matching (or assignment) algorithm, where motor neurons "compete" with each other to "win" muscle fibers. We show that this algorithm is simple to implement, theoretically sound, and effective in practice when evaluated on real-world bipartite matching problems. Thus, insights from the development of neural circuits can inform the design of algorithms for fundamental computational problems.

Optimal bipartite graph matching-based goal selection for policy-based hindsight learning

The sparse reward problem stands as a significant challenge in the field of reinforcement learning. Hindsight Experience Replay (HER) addresses this by goal relabeling, allowing the agent to learn from unsuccessful experiences. Some studies combine policy gradient methods with HER, resulting in policy-based hindsight learning algorithms. However, Policy-based hindsight learning involves the use of importance sampling, where the distribution of hindsight goals and the distribution of desired goals contribute to the computation of importance weights. When there is a significant difference between the two distributions, importance weights may become skewed, thereby impacting the evaluation of the policy and leading to suboptimal policies. To address this, we propose modeling the goal selection as an optimization problem for distribution matching. After we augment the original desired goals using Kernel Density Estimation (KDE), we further convert the optimization problem for distribution matching into a bipartite graph matching problem that minimizes the sum of weights. Our optimal bipartite graph matching-based hindsight goal selection method can select hindsight goals that are the most closely aligned with the original goals. Experimental results show that algorithms combined with the optimal bipartite graph matching-based hindsight goal selection outperform the original algorithms. Visualizations also demonstrate the superiority of our method in selecting hindsight goals.

Kartograf: A Geometrically Accurate Atom Mapper for Hybrid-Topology Relative Free Energy Calculations.

Relative binding free energy (RBFE) calculations have emerged as a powerful tool that supports ligand optimization in drug discovery. Despite many successes, the use of RBFEs can often be limited by automation problems, in particular, the setup of such calculations. Atom mapping algorithms are an essential component in setting up automatic large-scale hybrid-topology RBFE calculation campaigns. Traditional algorithms typically employ a 2D subgraph isomorphism solver (SIS) in order to estimate the maximum common substructure. SIS-based approaches can be limited by time-intensive operations and issues with capturing geometry-linked chemical properties, potentially leading to suboptimal solutions. To overcome these limitations, we have developed Kartograf, a geometric-graph-based algorithm that uses primarily the 3D coordinates of atoms to find a mapping between two ligands. In free energy approaches, the ligand conformations are usually derived from docking or other previous modeling approaches, giving the coordinates a certain importance. By considering the spatial relationships between atoms related to the molecule coordinates, our algorithm bypasses the computationally complex subgraph matching of SIS-based approaches and reduces the problem to a much simpler bipartite graph matching problem. Moreover, Kartograf effectively circumvents typical mapping issues induced by molecule symmetry and stereoisomerism, making it a more robust approach for atom mapping from a geometric perspective. To validate our method, we calculated mappings with our novel approach using a diverse set of small molecules and used the mappings in relative hydration and binding free energy calculations. The comparison with two SIS-based algorithms showed that Kartograf offers a fast alternative approach. The code for Kartograf is freely available on GitHub (https://github.com/OpenFreeEnergy/kartograf). While developed for the OpenFE ecosystem, Kartograf can also be utilized as a standalone Python package.

Popular critical matchings in the many-to-many setting

We consider the many-to-many bipartite matching problem in the presence of two-sided preferences and two-sided lower quotas. The input to our problem is a bipartite graph G=(A∪B,E), where each vertex in A∪B specifies a strict preference ordering over its neighbours. Each vertex has an upper quota denoting the maximum number of vertices that can be assigned to it. In addition, each vertex has a lower quota denoting the minimum number of vertices that need to be assigned to it. In the many-to-many setting with two-sided lower quotas, informally, a critical matching is a matching which fulfils vertex lower quotas to the maximum possible extent. This is a natural generalization of the definition of critical matching in the one-to-one setting considered by Kavitha (FSTTCS 2021). In this work, our goal is to find a popular matching in the set of critical matchings. A matching is popular in a given set of matchings if it remains undefeated in a head-to-head election with any matching in that set. Here, vertices cast votes between pairs of matchings. We show that there always exists a matching that is popular in the set of critical matchings. We present an efficient algorithm to compute such a matching of the largest size. We prove the popularity of our matching using a dual certificate.

Optimal transport methods for combinatorial optimization over two random point sets

We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} where the edge cost between two points is given by a pth power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d≥3,1≤p<d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3, 1\\le p<d$$\\end{document}. Previous results were limited to the range p<d/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<d/2$$\\end{document}. Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.

Online Bipartite Matching with Reusable Resources

We study the classic online bipartite matching problem with a twist: off-line vertices, called resources, are reusable. In particular, when a resource is matched to an online vertex, it is unavailable for a deterministic time duration d, after which it becomes available again for a rematch. Thus, a resource can be matched to many different online vertices over a period of time. Whereas recent work on the problem has resolved the asymptotic case in which we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of unit inventory and give the first algorithms that are provably better than the naïve greedy approach, which has a competitive ratio of (exactly) 0.5. Our first algorithm, which achieves a competitive ratio of 0.589, generalizes the classic RANKING algorithm for online bipartite matching of nonreusable resources (Karp et al. 1990) by reranking resources independently over time. Whereas reranking resources frequently has the same worst case performance as greedy, we show that reranking intermittently on a periodic schedule succeeds in addressing reusability of resources and performs significantly better than greedy in the worst case. Our second algorithm, which achieves a competitive ratio of 0.505, is a primal-dual randomized algorithm that works by suggesting up to two resources as candidate matches for every online vertex and then breaking the tie to make the final matching selection in a randomized correlated fashion over time. As a key component of our algorithm, we suitably adapt and extend the powerful technique of online correlated selection (Fahrbach et al. 2020) to reusable resources in order to induce negative correlation in our tie-breaking step and beat the competitive ratio of 0.5. Both of our results also extend to the case in which off-line vertices have weights. Funding: R. Niazadeh’s research is supported by the Asness Junior Faculty Fellowship at The University of Chicago Booth School of Business.

Fast primal-dual update against local weight update in linear assignment problem and its application

We consider a dynamic situation in the weighted bipartite matching problem: edge weights in the input graph are repeatedly updated and we are asked to maintain an optimal matching at any moment. A trivial approach is to compute an optimal matching from scratch each time an update occurs. In this paper, we show that if each update occurs locally around a single vertex, then a single execution of Dijkstra's algorithm is sufficient to preserve optimality with the aid of a dual solution. As an application of our result, we provide a faster implementation of the envy-cycle procedure for finding an envy-free allocation of indivisible items. Our algorithm runs in O(mn2) time, while the known bound of the original one is O(mn3), where n and m denote the numbers of agents and items, respectively.

Time-Efficient Joint UAV-BS Deployment and User Association Based on Machine Learning

In this paper, a time-efficient mechanism is proposed to solve the joint unmanned aerial vehicle (UAV) base station deployment and user/sensor association (UDUA) problem aiming at maximizing the downlink sum transmission throughput and reducing the time of on-line computations. In this work, two relevant sub-problems are decoupled from the joint UDUA problem: the first one is denoted as the user association sub-problem, for certain UAV base station (UAV-BS) positions, this sub-problem is settled for finding the strategy which matches aerial and ground nodes optimally. The second sub-problem is the positioning of UAV-BSs in order to obtain the best possible solution to the user association sub-problem from all possible positioning combinations for the UAV-BSs. In the proposed mechanism, the Kuhn-Munkres algorithm is used to solve the first sub-problem as an equivalent bipartite matching problem. For the UAV-BS deployment sub-problem, when the two user distributions own a high similarity, we theoretically prove that little performance decline will be introduced when the new user distribution’s optimal strategy is compared with choosing the optimal UAV-BS deployment strategy of stored user distributions. Based on our mathematical analyses, the similarity level between user distributions is well defined and becomes the key to solve the second sub-problem. According to experimental findings, the proposed UDUA mechanism, when compared to benchmark approaches, can provide near-optimum system performance with regard to average downlink total transmission throughput and failure rate with significantly decreased computing time.

Users' grouping algorithm for fairness improvement of NOMA‐based multi‐beams satellite networks intended for 5G

AbstractNon‐Orthogonal Multiple‐Access (NOMA)‐based Multibeam Satellite Networks (MBSNs) are intended for 5G implementation, and 5G networks set the requirement for both high system capacity as well as high system reliability to all users, which is achieved via high system's fairness. The NOMA network requires a user grouping algorithm (UGA), and most existing UGAs for 2‐users NOMA‐MBSNs, favor the system's capacity maximization with little regard to the system's fairness. Thus, this paper proposes a UGA that maximizes the fairness of 2‐users NOMA‐MBSNs. The algorithm seeks to group 2M users into M‐pairs satisfying three requirements, (i) a minimum channel‐gain margin; (ii) high channel‐correlation coefficients; and (iii) there is high fairness amongst the resulting pairs, in term of their respective channel‐correlation coefficients, to increase the users' fairness of the system. To achieve this, the 2M users are split into two sets of M‐far and M‐near users based on users' channel gains for pairing far and near users with high channel‐correlation coefficients. It will turn it into a maximization Bipartite Matching problem (BPMP) of channel‐correlation coefficients. The BPMP is addressed by means of the Hungarian Method (HM) to ensure high fairness amongst resulting pairs regarding their channel‐correlation coefficients. The proposed UGA demonstrates superior fairness compared to existing UGAs.

Generalized Category Discovery with Decoupled Prototypical Network

Generalized Category Discovery (GCD) aims to recognize both known and novel categories from a set of unlabeled data, based on another dataset labeled with only known categories. Without considering differences between known and novel categories, current methods learn about them in a coupled manner, which can hurt model's generalization and discriminative ability. Furthermore, the coupled training approach prevents these models transferring category-specific knowledge explicitly from labeled data to unlabeled data, which can lose high-level semantic information and impair model performance. To mitigate above limitations, we present a novel model called Decoupled Prototypical Network (DPN). By formulating a bipartite matching problem for category prototypes, DPN can not only decouple known and novel categories to achieve different training targets effectively, but also align known categories in labeled and unlabeled data to transfer category-specific knowledge explicitly and capture high-level semantics. Furthermore, DPN can learn more discriminative features for both known and novel categories through our proposed Semantic-aware Prototypical Learning (SPL). Besides capturing meaningful semantic information, SPL can also alleviate the noise of hard pseudo labels through semantic-weighted soft assignment. Extensive experiments show that DPN outperforms state-of-the-art models by a large margin on all evaluation metrics across multiple benchmark datasets. Code and data are available at https://github.com/Lackel/DPN.

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>

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.

3D Multi-Object Tracking With Adaptive Cubature Kalman Filter for Autonomous Driving

A crucial and challenging issue in autonomous driving is dynamic road environment detection and 3D multi object tracking. In this article, we propose a novel framework for online 3D multi-object tracking to eliminate the influence of inherent uncertainty and unknown biases in point cloud. A constant turn rate and velocity (CTRV) motion model is employed to estimate the future motion state, which are smoothed by a cubature Kalman filter (CKF) algorithm. A new affinity model is introduced to evaluate the similarity between trajectories and candidate detections for accurate and reliable data association which can be formulated as a bipartite matching problem. An adaptive cubature Kalman filter (ACKF) is given to remove the influence of unknown bias and to robustly update the tracked state. Accuracy and speed of the proposed tracking method are evaluated on the KITTI 3D multi-object tracking dataset, showing superior performance than the baselines.

Greediness is not always a vice: Efficient Discovery Algorithms for Assignment Problems

Finding a maximum-weight matching is a classical and well-studied problem in computer science, solvable in cubic time in general graphs. We introduce and consider in this work the “discovery” variant of the bipartite matching problem (or assignment problem) where edge weights are not provided as input but must be queried, requiring additional and costly computations. Hence, discovery algorithms are developed aiming to minimize the number of queried weights while providing guarantees on the computed solution. We show in this work the hardness of the underlying problem in general while providing several efficient algorithms that can make use of natural assumptions about the order in which the nodes are processed by the greedy algorithms. Our motivations for exploring this problem stem from finding practical solutions to maximum-weight matching in hypergraphs, a problem recently emerging in the formation of peer-to-peer energy sharing communities.

Voting-based efficient cluster ensemble fusion

Voting-based consensus clustering is a subset of consensus techniques that makes clear the cluster label mismatch issue. Finding the best relabeling for a given partition in relation to a reference partition is known as the voting problem. As a weighted bipartite matching problem, it is frequently formulated. We propose a more generic formulation of the voting problem as a regression problem with various and multiple-input variables in this work. We demonstrate how a recently developed cumulative voting system is an exception that corresponds to a linear regression technique. We employ a randomised ensemble creation method in which an excess of clusters are randomly chosen for each ensemble partition. In order to extract the consensus clustering from the combined ensemble representation and to calculate the number of clusters, we use an information-theoretic approach. Together with bipartite matching and cumulative voting, we use it. We provide empirical data demonstrating significant enhancements in clustering stability, estimation of the real number of clusters, and accuracy of clustering based on cumulative voting. The gains are made in comparison to recent consensus algorithms as well as bipartite matching-based consensus algorithms, which struggle with the selected ensemble generation technique.

Exploring science-technology linkages: A deep learning-empowered solution

In-depth exploration of the knowledge linkages between science and technology (S&T) is an essential prerequisite for accurately understanding the S&T innovation laws, promoting the transformation of scientific outcomes, and optimizing S&T innovation policies. A novel deep learning-based methodology is proposed to investigate S&T linkages, where papers and patents are applied to represent science and technology. In order to accurately and comprehensively reveal the linkages between science and technology topics, the proposed framework combines the information of knowledge structure with textual semantics. Furthermore, the exploration analysis is also conducted from the perspective of realizing the optimal matching between science and technology topics, which can realize combinatorial optimization of the S&T knowledge systems. Specifically, science and technology networks are constructed based on Node2Vec and BERT. Then, science and technology topics are identified based on the Fast Unfolding algorithm and Z-Score index. Finally, a science-technology bipartite graph is constructed, the S&T topic linkages identification task is successfully transferred into a bipartite matching problem, and the maximum-weight matching is identified using a Kuhn-Munkres bipartite algorithm. An experiment on natural language processing demonstrates the feasibility and reliability of the proposed methodology.

Stackelberg packing games

Stackelberg pricing games are pricing problems over a set of items. One player, the leader, sets prices for the items and the second player, the follower, buys a subset of items at minimal total cost subject to feasibility constraints. The constraints are determined by an optimization problem. For example, the Stackelberg shortest path game is used to optimize the income from road tolls (cf. [21]). The items are edges in a network graph and the follower buys a subset of edges forming a path.The Stackelberg pricing games studied in the literature are formulated on top of covering problems. In this paper, we introduce pricing games based on packing problems. We are interested in the complexity of computing leader-optimal prices depending on different types of follower-constraints. We show that optimal prices can be computed in polynomial time if the follower-constraints are determined by the well-known interval scheduling problem. This problem is equivalent to the independent set problem on interval graphs. In case the follower-constraints are based on the independent set problem on perfect graphs or on the bipartite matching problem, we prove APX-hardness for the pricing problem.On a more general note, we prove Σ2p-completeness if the follower-constraints are given by a packing problem that is NP-complete, i.e., the leader's pricing problem is hard even if she has an NP-oracle at hand.

Fast Proactive Repair in Erasure-Coded Storage: Analysis, Design, and Implementation

Erasure coding offers a storage-efficient redundancy mechanism for maintaining data availability guarantees in large-scale storage clusters, yet it also incurs high performance overhead in failure repair. Recent developments in accurate disk failure prediction allow soon-to-fail (STF) nodes to be repaired in advance, thereby opening new opportunities for accelerating failure repair in erasure-coded storage. To this end, we present a fast proactive repair solution called FastPR, which carefully couples two repair methods, namely migration (i.e., relocating the chunks of an STF node) and reconstruction (i.e., decoding the chunks of an STF node through erasure coding), so as to fully parallelize the repair operation across the storage cluster. FastPR solves a bipartite maximum matching problem and schedules both migration and reconstruction in a parallel fashion. We show that FastPR significantly reduces the repair time over the baseline repair approaches for both Reed-Solomon codes and Azures Local Reconstruction Codes via mathematical analysis, large-scale simulation, and Amazon EC2 experiments.

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.