Bipartite Graph Matching Problem Research Articles

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.

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.

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.

Multi-task dispatch of shared autonomous electric vehicles for Mobility-on-Demand services – combination of deep reinforcement learning and combinatorial optimization method

The Autonomous Mobility-on-Demand system is an emerging green and sustainable transportation system providing on-demand mobility services for urban residents. To achieve the best recharging, delivering, and repositioning task assignment decision-making process for shared autonomous electric vehicles, this paper formulates the fleet dynamic operating process into a multi-agent multi-task dynamic dispatching problem based on Markov Decision Process. Specifically, the decision-making process at each time step is divided into 3 sub-processes, among which recharging and delivery task assignment processes are transformed into a maximum weight matching problem of bipartite graph respectively, and the repositioning task assignment process is quantified as a maximum flow problem. Kuhn-Munkres Algorithm and Edmond-Karp Algorithm are adopted to solve the above two mathematical problems to achieve the optimal task allocation policy. To further improve the dispatching performance, a new instant reward function balancing order income with trip satisfaction is designed, and a state-value function estimated by Back Propagation-Deep Neural Network is defined as a matching degree between each shared autonomous electric vehicle and each delivery task. The numerical results show that: (i) a reward function focusing on income and satisfaction can increase total revenue by 33.2%, (ii) the introduction of task allocation repositioning increases total revenue by 50.0%, (iii) a re-estimated state value function leads to a 2.8% increase in total revenue, (iv) the combination of charging and task repositioning can reduce user waiting time and significantly improve user satisfaction with the trip.

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.

LPTD: a novel linear programming-based topology determination method for cryo-EM maps.

Topology determination is one of the most important intermediate steps toward building the atomic structure of proteins from their medium-resolution cryo-electron microscopy (cryo-EM) map. The main goal in the topology determination is to identify correct matches (i.e. assignment and direction) between secondary structure elements (SSEs) (α-helices and β-sheets) detected in a protein sequence and cryo-EM density map. Despite many recent advances in molecular biology technologies, the problem remains a challenging issue. To overcome the problem, this article proposes a linear programming-based topology determination (LPTD) method to solve the secondary structure topology problem in three-dimensional geometrical space. Through modeling of the protein's sequence with the aid of extracting highly reliable features and a distance-based scoring function, the secondary structure matching problem is transformed into a complete weighted bipartite graph matching problem. Subsequently, an algorithm based on linear programming is developed as a decision-making strategy to extract the true topology (native topology) between all possible topologies. The proposed automatic framework is verified using 12 experimental and 15 simulated α-β proteins. Results demonstrate that LPTD is highly efficient and extremely fast in such a way that for 77% of cases in the dataset, the native topology has been detected in the first rank topology in <2 s. Besides, this method is able to successfully handle large complex proteins with as many as 65 SSEs. Such a large number of SSEs have never been solved with current tools/methods. The LPTD package (source code and data) is publicly available at https://github.com/B-Behkamal/LPTD. Moreover, two test samples as well as the instruction of utilizing the graphical user interface have been provided in the shared readme file. Supplementary data are available at Bioinformatics online.

Auction-Based VM Allocation for Deadline-Sensitive Tasks in Distributed Edge Cloud

Edge cloud computing is a new paradigm in which the computation and storage services of remote cloud data centers are moved to Edge Cloud Nodes (ECNs) in network edges. Compared to traditional cloud data centers, ECNs are geographically close to mobile users so the communication latency is significantly reduced. In this paper, we study the problem of allocating Virtual Machine (VM) resources in geo-distributed ECNs to mobile users by using the auction theory. First, we treat mobile users and ECNs as the buyers and sellers of the VM resource auction, respectively. Then, we model the VM resource allocation problem as an <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -to-one weighted bipartite graph matching problem with 0-1 knapsack constraints. Since this problem is NP-hard, we design a greedy approximation algorithm to determine the winners of the auction, based on which we propose a truthful Auction-based VM resource Allocation (AVA) mechanism to solve the problem. Moreover, we prove that the AVA mechanism not only achieves an approximately optimal solution for winner selection, but also has the properties of truthfulness, individual rationality, and computational efficiency. Finally, we conduct extensive simulations on real traces to verify the significant performances of the proposed AVA mechanism.

Crater Detection and Recognition Method for Pose Estimation

A crater detection and recognition algorithm is the key to pose estimation based on craters. Due to the changing viewing angle and varying height, the crater is imaged as an ellipse and the scale changes in the landing camera. In this paper, a robust and efficient crater detection and recognition algorithm for fusing the information of sequence images for pose estimation is designed, which can be used in both flying in orbit around and landing phases. Our method consists of two stages: stage 1 for crater detection and stage 2 for crater recognition. In stage 1, a single-stage network with dense anchor points (dense point crater detection network, DPCDN) is conducive to dealing with multi-scale craters, especially small and dense crater scenes. The fast feature-extraction layer (FEL) of the network improves detection speed and reduces network parameters without losing accuracy. We comprehensively evaluate this method and present state-of-art detection performance on a Mars crater dataset. In stage 2, taking the encoded features and intersection over union (IOU) of craters as weights, we solve the weighted bipartite graph matching problem, which is matching craters in the image with the previously identified craters and the pre-established craters database. The former is called “frame-frame match”, or FFM, and the latter is called “frame-database match”, or FDM. Combining the FFM with FDM, the recognition speed is enabled to achieve real-time on the CPU (25 FPS) and the average recognition precision is 98.5%. Finally, the recognition result is used to estimate the pose using the perspective-n-point (PnP) algorithm and results show that the root mean square error (RMSE) of trajectories is less than 10 m and the angle error is less than 1.5 degrees.

Achieving arbitrary throughput-fairness trade-offs in the inter cell interference coordination with fixed transmit power problem

We study the problem of inter cell interference coordination (ICIC) with fixed transmit power in OFDMA-based cellular networks, in which each base station (BS) needs to decide as to which subchannel, if any, to allocate to each of its associated mobile stations (MS) for data transmission. In general, there exists a trade-off between the total throughput (sum of throughputs of all the MSs) and fairness under the allocations found by resource allocation schemes. We introduce the concept of \(\tau -\alpha -\)fairness by modifying the concept of \(\alpha -\)fairness, which was earlier proposed in the context of congestion control for packet-switched networks. The concept of \(\tau -\alpha -\)fairness allows us to achieve arbitrary trade-offs between total throughput and fairness by selecting an appropriate value of \(\alpha\) in \([0,\infty )\). We show that for every \(\alpha \in [0,\infty )\) and every \(\tau > 0\), the problem of finding a \(\tau -\alpha -\)fair allocation is NP-Complete. Further, we show that for every \(\alpha \in [0, \infty )\), there exist thresholds such that if the potential interference levels experienced by each MS on every subchannel are above the threshold values, then the problem can be optimally solved in polynomial time by reducing it to the bipartite graph matching problem. Also, we propose a simple, distributed subchannel allocation algorithm for the ICIC problem, which is flexible, requires a small amount of time to operate, and requires information exchange among only neighboring BSs. We investigate via simulations as to how the algorithm parameters should be selected so as to achieve any desired trade-off between the total throughput and fairness. Finally, we compare the performance of the proposed algorithm with those of the simulated annealing based and opportunistic subchannel allocation algorithms in terms of the total throughput, fairness index and computational complexity.

UAV-Aided Cooperative Data Collection Scheme for Ocean Monitoring Networks

In this article, we present an unmanned aerial vehicle (UAV)-aided ocean monitoring network for remote oceanic data collection, in which monitoring data are transmitted first from battery-powered underwater sensor nodes (USNs) to sea surface sink nodes (SNs) in a data collection cycle using underwater acoustic communication, and then a UAV hovering in air collects all the data from SNs and relays them to a ground base station via wireless communication links. Aiming at maximizing network lifetime, we model the resource allocation, USN-to-SN access, and SN-to-UAV access issues as a mixed-integer nonconvex optimization problem. To efficiently solve it, we decompose the optimization into two stages. The first stage is to minimize time consumption in an SN-to-UAV nonorthogonal multiple access process and we solve it by designing a UAV deployment scheme, a subchannel matching scheme, and a joint power and time allocation scheme, based on which, the second stage is to maximize the residual energies of USNs in USN-to-SN transmissions under a modified frequency-division multiple access strategy in each collection cycle. The second-stage optimization is further decomposed into some similar subproblems, and each of them is considered as a bipartite graph matching problem between USNs and underwater acoustic channels. For each subproblem, we propose improved weight-based matching and bisection-based searching algorithms. Finally, we design a low-complexity iteration algorithm to approximate the optimal solution of the original problem by solving these subproblems. The simulation results validate the effectiveness of our proposals.

On the induced matching problem in hamiltonian bipartite graphs

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

Solving the α-helix correspondence problem at medium-resolution Cryo-EM maps through modeling and 3D matching

Cryo-electron microscopy (cryo-EM) has recently emerged as a prominent biophysical method for macromolecular structure determination. Many research efforts have been devoted to produce cryo-EM images, density maps, at near-atomic resolution. Despite many advances in technology, the resolution of the generated density maps may not be sufficiently adequate and informative to directly construct the atomic structure of proteins. At medium-resolution (∼4-10 Å), secondary structure elements (α-helices and β-sheets) are discernible, whereas finding the correspondence of secondary structure elements detected in the density map with those on the sequence remains a challenging problem. In this paper, an automatic framework is proposed to solve α-helix correspondence problem in three-dimensional space. Through modeling of the sequence with the aid of a novel strategy, the α-helix correspondence problem is initially transformed into a complete weighted bipartite graph matching problem. An innovative correlation-based scoring function based on a well-known and robust statistical method is proposed for weighting the graph. Moreover, two local optimization algorithms, which are Greedy and Improved Greedy algorithms, have been presented to find α-helix correspondence. A widely used data set including 16 reconstructed and 4 experimental cryo-EM maps were chosen to verify the accuracy and reliability of the proposed automatic method. The experimental results demonstrate that the automatic method is highly efficient (86.25% accuracy), robust (11.3% error rate), fast (∼1.4 s), and works independently from cryo-EM skeleton.

Shape correspondence based on Kendall shape space and RAG for 2D animation

We introduce a 2D vectorized shape correspondence method based on Kendall shape space and region adjacency graph. We regard shape correspondence in 2D animation as a weighted bipartite graph matching problem and optimize it by means of various weights, including geometric attribute, local topological information, and global topological information in 2D animation drawing. The measuring method in Kendall shape space is introduced to determine similarity between two regions and a local adjacent region matching method is presented to associate the vicinity of the corresponding region by utilizing local topology information. Ultimately, we propose a globally optimal shape matching method, which exploits the global topological information to get the final result of the shape correspondence. Our approach can efficiently match associated regions that have exaggerated deformation and unstable topology between two shapes in 2D animation. It is also robust to the similarity transformation of shapes.

Resource allocation for virtual reality content sharing based on 5G D2D multicast communication

As the development of wireless virtual reality (VR), a great disparity exists among the huge content transmission, rigorous QoS guarantees and the limited bandwidth of cellular networks. With the increasing of cache capacity on user devices, direct content sharing between user devices is a promising solution to solve this problem. To meet the quality of service (QoS) requirement of wireless VR transmission, this paper proposes a content sharing scheme based on 5G device-to-device (D2D) Multicast Communication. In this way, some adjacent VR users can form multicasting clustering to share VR content by working on D2D communication mode. The VR D2D multicasting clusters reuse the uplink channel resource of ordinary VR cellular users in the same cell. The basic prerequisite is that the degrading on QoS of these ordinary VR cellular users can be affordable. We design a two-step scheme to solve this radio resource allocation problem for VR content sharing. Firstly, we find out the optimal transmitting power for each VR user devices by geometric proximity, which metrics are affected by wireless VR throughput, tracking accuracy and delay. In the second step, we transform the channel allocation problem into a bipartite graph matching problem based on the transmitting power metrics, which is optimally solved by the Hungarian algorithm. The simulation results show that the VR content sharing based on 5G D2D communication technology can achieve larger system throughput gain and lower transmission delay. Compared with heuristic scheme and stochastic scheme, the proposed scheme can increase the throughput of the overall network by about 50% and 12%, respectively.

SRA: Secure Reverse Auction for Task Assignment in Spatial Crowdsourcing

In this paper, we study a new type of spatial crowdsourcing, namely competitive detour tasking, where workers can make detours from their original travel paths to perform multiple tasks, and each worker is allowed to compete for preferred tasks by strategically claiming his/her detour costs. The objective is to make suitable task assignment by maximizing the social welfare of crowdsourcing systems and protecting workers’ private sensitive information. We first model the task assignment problem as a reverse auction process. We formalize the winning bid selection of reverse auction as an $n$ n -to-one weighted bipartite graph matching problem with multiple 0-1 knapsack constraints. Since this problem is NP-hard, we design an approximation algorithm to select winning bids and determine corresponding payments. Based on this, a Secure Reverse Auction (SRA) protocol is proposed for this novel spatial crowdsourcing. We analyze the approximation performance of the proposed protocol and prove that it has some desired properties, including truthfulness, individual rationality, computational efficiency, and security. To the best of our knowledge, this is the first theoretically provable secure auction protocol for spatial crowdsourcing systems. In addition, we also conduct extensive simulations on a real trace to verify the performance of the proposed protocol.

Enabling Security-Aware D2D Spectrum Resource Sharing for Connected Autonomous Vehicles

With the emergence of automated driving technology, wireless demand for secure information exchange among automated vehicles has increased dramatically. To this end, we design a security-aware dynamic device-to-device (D2D) spectrum resource sharing mechanism to enhance the security of vehicular D2D communications with the improved spectrum efficiency. Considering both resource block (RB) sharing and power control, we model this joint D2D spectrum resource sharing process as a weighted bipartite graph matching problem whose weights are obtained through deriving the closed-form solutions of power control in an algebraic method. Then, the global optimal solutions of the matching problem are obtained by using the Hungarian algorithm. Furthermore, a security-aware RB and power allocation (SA-RBPA) mechanism compatible with existing cellular networks is proposed for the small cell base station applications. Extensive simulation results have shown that, compared with existing approaches that consider RB reusing strategy or power control scheme optimization alone, the SA-RBPA scheme is able to realize a better spectrum efficiency and security performance.

Reducing Invertible Embedding Distortion Using Graph Matching Model

Invertible embedding allows the original cover and embedded data to be perfectly reconstructed. Conventional methods use a well-designed predictor and fully exploit the carrier characteristics. Due to the diversity, it is actually hard to accurately model arbitrary covers, which limits the practical use of methods relying heavily on content characteristics. It has motivated us to revisit invertible embedding operations and propose a general graph matching model to generalize them and further reduce the embedding distortion. In the model, the rate-distortion optimization task of invertible embedding is derived as a weighted bipartite graph matching problem. In the bipartite graph, the nodes represent the values of cover elements, and the edges indicate the candidate modifications. Each edge is associated with a weight indicating the corresponding embedding distortion for the connected nodes. By solving the minimum weight maximum matching problem, we can find the optimal embedding strategy under the constraint. Since the proposed work is a general model, it can be incorporated into existing works to improve their performance, or used for designing new invertible embedding systems. We incorporate the proposed work into a part of state-of-the-arts, and experiments show that it significantly improves the rate-distortion performance. To the best knowledge of the authors, it is probably the first work studying rate-distortion optimization of invertible embedding from the perspective of graph matching model.