Bipartite Matching Problem Research Articles

Multi-Agent DRL-Based Hungarian Algorithm (MADRLHA) for Task Offloading in Multi-Access Edge Computing Internet of Vehicles (IoVs)

This paper investigates the computation offloading problem in a high mobility internet of vehicles (IoVs) environment, aiming to guarantee latency, energy consumption, and payment cost requirements. Both moving and parked vehicles are utilized as fog nodes. Vehicles in high mobility environments need collaborative interactions in a decentralized manner for better network performances, where agent action space grows exponentially with the number of vehicles. The vehicular mobility introduces additional dynamicity in the network, and the learning agent requires a joint cooperative behavior for establishing convergence. The traditional deep reinforcement learning (DRL)-based offloading in IoV ignores other agent’s actions during the training process as an independent learner, which makes a lack of robustness against the high mobility environment. To overcome it, we develop a cooperative three-layer, more generic decentralized vehicle-assisted multi-access edge computing (VMEC) network, where vehicles in associated RSU and neighbor RSUs are in the bottom fog layer, MEC servers are in the middle cloudlet layer, and cloud in the top layer. Then multi-agent DRL-based Hungarian algorithm (MADRLHA) in the bipartite graph maximum matching problem is applied to solve dynamic task offloading in VMEC. Extensive experimental results and comprehensive comparisons are conducted to illustrate the superiority of our proposed method.

Joint Resource Allocation Optimization for the Wireless Backhaul Link of User-Centric Cell-Free NOMA Networks

In this paper, we investigate the downlink resource allocation problem for the user-centric ultra-dense networks (UUDN) which can provide high area throughput density. UUDN can effectively improve the user experience since one user can be served by more than one access points (APs) simultaneously. However, the ultra-dense node deployment will also result in serious co-channel interference which will degrade the performance of the networks. Nonorthogonal multiple access (NOMA) is introduced to reduce the interference caused by the network densification. We consider the resource allocation for both the wireless backhaul downlink and the user access downlink at the same time in UUDN with NOMA, which involves the APs grouping and power allocation. The resource allocation problem is formulated as a mixed integer nonconvex optimization problem due to the combinatorial nature of APs grouping. We first decompose the problem into the APs grouping subproblem and power allocation subproblem to reduce the computational complexity. For the APs grouping subproblem, we model it as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. When the APs grouping is fixed, the original problem is still difficult to tackle since the power allocation is a max-min optimization problem. The KKT conditions are applied to optimize the power allocation subproblem. Simulation results illustrate that the proposed resource allocation algorithm can effectively improve the system throughput of UUDN.

A nonasymptotic analysis for re‐solving heuristic in online matching

We investigate an online edge‐weighted bipartite matching problem with general capacity constraints. In this problem, the resources are offline and nonreplenishable with different capacities. Demands arrive online and each requests a certain amount of resources. The goal is to maximize the reward generated by successful matches. We model the offline optimization problem as a deterministic linear program (LP) and present multiple randomized online algorithms based on the solution to the offline LP. We analyze the performance guarantee of each algorithm in terms of its competitive ratio (CR). Importantly, we introduce a re‐solving heuristic that periodically recomputes the offline LP and uses the updated offline solution to guide the online algorithm decisions. We find that the algorithm's CR can be significantly improved when re‐solving at carefully selected time steps. Finally, we investigate the value of the demand distribution in further improving the algorithm efficiency. We conduct extensive numerical studies to demonstrate the efficiency of the proposed algorithms. The effect of market conditions on the algorithm performance is also investigated.

Secretary Matching with Vertex Arrivals and No Rejections

Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in a uniformly random order and the edge-weights are arbitrary positive numbers. For the capacitated online bipartite matching problem in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the online general (non-bipartite) matching problem, where all vertices arrive online, we give a 3.34-competitive algorithm. We also study the online roommate matching problem, in which each room (offline vertex) holds 2 persons (online vertices). Persons derive non-negative additive utilities from their room as well as roommate. In this model, with the goal of maximizing the sum of utilities of all persons, we give a 7.96-competitive algorithm. This is an improvement over the 24.72 approximation factor in prior work.

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.

Matching-based capture strategies for 3D heterogeneous multiplayer reach-avoid differential games

This paper studies a 3D multiplayer reach-avoid differential game with a goal region and a play region. Multiple pursuers defend the goal region by consecutively capturing multiple evaders in the play region. The players have heterogeneous moving speeds and the pursuers have heterogeneous capture radii. Since this game is hard to analyze directly, we decompose the whole game as many subgames which involve multiple pursuers and only one evader. Then, these subgames are used as a building block for the pursuer–evader matching. First, for multiple pursuers and one evader, we introduce an evasion space (ES) method characterized by a potential function to construct a guaranteed pursuer winning strategy. Then, based on this strategy, we develop conditions to determine whether a pursuit team can guard the goal region against one evader. It is shown that in 3D, if a pursuit team is able to defend the goal region against an evader, then at most three pursuers in the team are necessarily needed. We also compute the value function of the Hamilton–Jacobi–Isaacs (HJI) equation for a special subgame of degree. To capture the maximum number of evaders in the open-loop sense, we formulate a maximum bipartite matching problem with conflict graph (MBMC). We show that the MBMC is NP-hard and design a polynomial-time constant-factor approximation algorithm to solve it. Finally, we propose a receding horizon strategy for the pursuit team where in each horizon an MBMC is solved and the strategies of the pursuers are given. We also extend our results to the case of a bounded convex play region where the evaders escape through an exit. Two numerical examples are provided to demonstrate the results.

Online Learning Bipartite Matching with Non-stationary Distributions

Online bipartite matching has attracted wide interest since it can successfully model the popular online car-hailing problem and sharing economy. Existing works consider this problem under either adversary setting or i.i.d. setting. The former is too pessimistic to improve the performance in the general case; the latter is too optimistic to deal with the varying distribution of vertices. In this article, we initiate the study of the non-stationary online bipartite matching problem, which allows the distribution of vertices to vary with time and is more practical. We divide the non-stationary online bipartite matching problem into two subproblems, the matching problem and the selecting problem, and solve them individually. Combining Batch algorithms and deep Q-learning networks, we first construct a candidate algorithm set to solve the matching problem. For the selecting problem, we use a classical online learning algorithm, Exp3, as a selector algorithm and derive a theoretical bound. We further propose CDUCB as a selector algorithm by integrating distribution change detection into UCB. Rigorous theoretical analysis demonstrates that the performance of our proposed algorithms is no worse than that of any candidate algorithms in terms of competitive ratio. Finally, extensive experiments show that our proposed algorithms have much higher performance for the non-stationary online bipartite matching problem comparing to the state-of-the-art.

Distributionally robust scheduling algorithms for total flow time minimization on parallel machines using norm regularizations

In this paper, we study a distributionally robust parallel machines scheduling problem, minimizing the total flow time criterion. The distribution of uncertain processing times is subject to ambiguity belonging to a set of distributions with constrained mean and covariance. We show that the problem can be cast as a deterministic optimization problem, with the objective function composed of an expectation and a regularization term given as an ℓp norm. The main question we ask and answer is whether the particular choice of the used ℓp norm affects the computational complexity of the problem and the robustness of its solution. We prove that if durations of the jobs are independent, the solution in terms of any ℓp norm can be solved in a pseudopolynomial time, by the reduction to a non-linear bipartite matching problem. We also show an efficient, polynomial-time algorithm for ℓ1 case. Furthermore, for instances with dependent durations of the jobs, we propose computationally efficient formulation and an algorithm that uses ℓ1 norm. Moreover, we identify a class of covariance matrices admitting a faster, polynomial-time algorithm. The computational experiments show that the proposed algorithms provide solutions with a similar quality to the existing algorithms while having significantly better computational complexities.

Low-Latency Federated Learning Over Wireless Channels With Differential Privacy

In federated learning (FL), model training is distributed over clients and local models are aggregated by a central server. The performance of uploaded models in such situations can vary widely due to imbalanced data distributions, potential demands on privacy protections, and quality of transmissions. In this paper, we aim to minimize FL training delay over wireless channels, constrained by overall training performance as well as each client’s differential privacy (DP) requirement. We solve this problem in a multi-agent multi-armed bandit (MAMAB) framework to deal with the situation where there are multiple clients confronting different unknown transmission environments, e.g., channel fading and interference. Specifically, we first transform long-term constraints on both training performance and each client’s DP into a virtual queue based on the Lyapunov drift technique. Then, we convert the MAMAB to a max-min bipartite matching problem at each communication round, by estimating rewards with the upper confidence bound (UCB) approach. More importantly, we propose two efficient solutions to this matching problem, i.e., a modified Hungarian algorithm and greedy matching with a better alternative (GMBA), of which the former can achieve the optimal solution with high complexity while the latter approaches a better trade-off by enabling verified low-complexity with little performance loss. In addition, we develop an upper bound on the expected regret of this MAMAB based FL framework, which shows a linear growth over the logarithm of communication rounds, justifying its theoretical feasibility. Extensive experimental results are conducted to validate the effectiveness of our proposed algorithms, and the impacts of various parameters on the FL performance over wireless edge networks are also discussed.

A tight bound for shortest augmenting paths on trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree T=(W⊎B,E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. It was conjectured by Chaudhuri et al. [6] that the total length of all shortest augmenting paths found is O(nlog⁡n). In this paper we prove a tight O(nlog⁡n) upper bound for the total length of shortest augmenting paths for trees improving over O(nlog2⁡n) bound [3].

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.

Achieving Extremely Low-Latency in Industrial Internet of Things: Joint Finite Blocklength Coding, Resource Block Matching, and Performance Analysis

To enable a number of emerging applications, efforts from industry and academia have started to focus on defining 6G systems, in which more stringent requirements than those imposed on 5G systems are being considered. In particular, some 6G applications may require extremely low-latency on the order of 0.1ms, through which practical designs of channel coding can be investigated based on Finite-Blocklength Coding (FBC). In this paper, we focus on a joint FBC scheme over multi-user downlinks in the Industrial Internet of Things (IIoT), in which only several symbol durations are available for users within a requirement of extremely low-latency. Since a higher coding rate is obtained by enlarging the blocklength of FBC, we jointly encode users’ data bits over their allocated resources, through which an enlarged blocklength is attained. Specifically, we first formulate the multi-user joint encoding design with a matrix-based method. Then, we present the optimal power-constrained throughput within the extremely low-latency requirement by formulating a nonlinear bipartite matching problem. We finally demonstrate the benefit resulting from the joint FBC in terms of each user’s maximum obtainable distance. With the distance to each user varying, we also perform an analysis of the variation of the optimal power-constrained throughput.

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.

(Un)stable matchings with blocking costs

We introduce and study weighted bipartite matching problems under strict preferences where blocking edges can be paid for, thus imposing costs rather than constraints as in classical models. We focus on the setting in which the weight of an edge represents the benefit from including it in the matching and/or the cost if it is a blocking edge. We show that this setting encompasses interesting special cases that remain polynomially-solvable, although it becomes APX-hard even in a quite restricted case.

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.

A combinatorial algorithm for computing the rank of a generic partitioned matrix with $$2 \times 2$$ submatrices

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{\alpha \beta} x_{\alpha \beta})$, where $A_{\alpha \beta}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{\alpha \beta}$ is an indeterminate for $\alpha = 1,2,\dots, \mu$ and $\beta = 1,2, \dots, \nu$. This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018) and Garg, Gurvits, Oliveiva, and Wigderson (2019), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial $O((\mu \nu)^2 \min \{ \mu, \nu \})$-time algorithm for computing the symbolic rank of a $(2 \times 2)$-type generic partitioned matrix of size $2\mu \times 2\nu$. Our algorithm is inspired by the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of $A$ for an arbitrary field $\mathbf{F}$.

Optimizing matching time intervals for ride-hailing services using reinforcement learning

Matching trip requests and available drivers efficiently is considered a central operational problem for ride-hailing services. A widely adopted matching strategy is to accumulate a batch of potential passenger-driver matches and solve bipartite matching problems repeatedly. The efficiency of matching can be improved substantially if the matching is delayed by adaptively adjusting the matching time interval. The optimal delayed matching is subject to the trade-off between the delay penalty and the reduced wait cost and is dependent on the system’s supply and demand states. Searching for the optimal delayed matching policy is challenging, as the current policy is compounded with past actions. To this end, we tailor a family of reinforcement learning-based methods to overcome the curse of dimensionality and sparse reward issues. In addition, this work provides a solution to spatial partitioning balance between the state representation error and the optimality gap of asynchronous matching. Lastly, we examine the proposed methods with real-world taxi trajectory data and garner managerial insights into the general delayed matching policies. The focus of this work is single-ride service due to limited access to shared ride data, while the general framework can be extended to the setting with a ride-pooling component.

Potential of carpool for network traffic management

This study examines the impact of carpool on network traffic in a highly idealized futuristic world, where all travelers are willing to participate in carpool arranged by a Transportation Network Company. We build a parsimonious carpool model that focuses on the trade-off between inconvenience costs and travel cost savings. Underlying the model is a nonlinear bipartite matching problem that seeks to maximize commuters’ welfare. By assuming the congestion effect is negligible, we derive several useful analytical results. When the inconvenience cost is less than the median trip valuation of a rider, the platform could always achieve an almost perfect match while maximizing commuters’ welfare, which corresponds to a 50% reduction in vehicular traffic flow. In the case of perfect match, if there is an even number of travelers, we propose a pricing policy that possesses all desired properties of the Vickrey-Clark-Groves (VCG) policy – a benchmark truthful policy for achieving socially optimal solution – but runs a lower deficit. Otherwise, we show the VCG policy always generates a profit. If the inconvenience cost is too high, the perfect match is no longer socially optimal, but the VCG policy still yields a positive profit. Solutions from numerical experiments generally agree with the analytical results. They also suggest that matching across O-D pairs occurs only when it has a significantly lower inconvenience cost than matching within, an unlikely event in reality. Moreover, when cross O-D matching does become prevalent, it leads to higher vehicle miles travelled, hence worse congestion. Thus, from the point of view of traffic management, cross O-D carpool should not be encouraged.

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.