Maximum Weight Matching Research Articles

A Semidefinite Programming-Based Branch-and-Cut Algorithm for Biclustering

Biclustering, also called co-clustering, block clustering, or two-way clustering, involves the simultaneous clustering of both the rows and the columns of a data matrix into distinct groups such that the rows and columns within a group display similar patterns. As a model problem for biclustering, we consider the k-densest disjoint biclique problem, whose goal is to identify k disjoint complete bipartite subgraphs (called bicliques) of a given weighted complete bipartite graph such that the sum of their densities is maximized. To address this problem, we present a tailored branch-and-cut algorithm. For the upper-bound routine, we consider a semidefinite programming relaxation and propose valid inequalities to strengthen the bound. We solve this relaxation in a cutting-plane fashion using a first-order method. For the lower bound, we design a maximum weight matching rounding procedure that exploits the solution of the relaxation solved at each node. Computational results on both synthetic and real-world instances show that the proposed algorithm can solve instances approximately 20 times larger than those handled by general-purpose solvers. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0683 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0683 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Improving ranking quality and fairness in Swiss-system chess tournaments

Abstract The International Chess Federation (FIDE) imposes a voluminous and complex set of player pairing criteria in Swiss-system chess tournaments and endorses computer programs that are able to calculate the prescribed pairings. The purpose of these formalities is to ensure that players are paired fairly during the tournament and that the final ranking corresponds to the players’ true strength order. We contest the official FIDE player pairing routine by presenting alternative pairing rules. These can be enforced by computing maximum weight matchings in a carefully designed graph. We demonstrate by extensive experiments that a tournament format using our mechanism (1) yields fairer pairings in the rounds of the tournament and (2) produces a final ranking that reflects the players’ true strengths better than the state-of-the-art FIDE pairing system.

Energy landscapes of some matching-problem ensembles

The maximum-weight matching problem and the behavior of its energy landscape is numerically investigated. We apply a perturbation method adapted from the analysis of spin glasses. This method provides insight into the complexity of the energy landscape of different ensembles. Erdős–Rényi graphs and ring graphs with randomly added edges are considered, and two types of distributions for the random edge weights are used. Fast and scalable algorithms exist for maximum weight matching, allowing us to study large graphs with more than 105 nodes. Our results show that the structure of the energy landscape for standard ensembles of matching is simple, comparable to the energy landscape of a ferromagnet. Nonetheless, for some of the ensembles presented here, our results allow for the presence of complex energy landscapes in the spirit of a replica-symmetry breaking scenario.

Blocking Trails for f-factors of Multigraphs

Blocking flows were introduced by Dinic (Soviet Math Doklady 11: 1277–1280, 1970) to speed up the computation of maximum network flows. They have been used in algorithms for problems such as maximum cardinality matching of bipartite graphs Hopcroft and Karp (SIAM J Comput 2(4), 225–231, 1973) and general graphs Micali and Vazirani (in: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, 17–27, 1980), maximum weight matching of general graphs Gabow and Tarjan (J ACM 38(4), 815–853, 1991), and many others. The blocking algorithm of Gabow and Tarjan (1991) for matching is based on depth-first search. We extend the depth-first search approach to find f-factors of general multigraphs. Here f is an arbitrary integral-valued function on vertices, an f-matching is a subgraph where every vertex x has degree $$\le f(x)$$ , an f-factor has equality in every degree bound. A set of blocking trails for an f-matching M is a maximal collection $$\mathcal{A}$$ of edge-disjoint augmenting trails such that $$M\bigoplus _{A\in {\mathcal{A}}} A $$ is a valid f-matching. Blocking trails are needed in efficient algorithms for maximum cardinality f-matching Huang and Pettie (Algorithmica 84(7): 1952–1992, 2022), maximum weight f-factors/matchings by scaling Duan et al. (In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), Vol. 168 of LIPIcs, 41:1-41:17, 2020; Gabow (A weight-scaling algorithm for f-factors of multigraphs. arXiv:2010.01102 , 2020), and approximate maximum weight f-factors and f-edge covers Huang and Pettie (2022). Since these algorithms find many sets of blocking trails, the time to find blocking trails is a dominant factor in the running time. Our blocking trail algorithm runs in linear time O(m). In independent work and using a different approach, Huang and Pettie (2022) present a blocking trails algorithm using time $$O(m\alpha (m,n))$$ . As examples of the time bounds for the above applications, an approximate maximum weight f-factor is found in time $$O(m\,\alpha (m,n))$$ using Huang and Pettie (2022), and our algorithm eliminates the factor $$\alpha (m,n)$$ . Similarly a maximum weight f-factor is found in time $$O(\sqrt{\Phi \log \Phi }\, m\,\alpha (m,n)\, \log (\Phi W))\,$$ using Huang and Pettie (2022) , ( $$\Phi =\sum _{v\in V} f(v)$$ , W the maximum edge weight) and our algorithm eliminates the $$\alpha (m,n)$$ factor, making the time within a factor $$\sqrt{\log {\Phi }}$$ of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that a fixed vertex can occur many times in a given search. This does not occur in ordinary matching or in algorithms for maximum cardinality or maximum weight f-matching. These multiple occurrences can create a new variant of blossom, the “skew blossom”. Also they can make blossoms become “incomplete”, i.e., partially processed yet still relevant in future searches.

A distributed message passing algorithm for computing perfect demand matching

In this paper, we consider the perfect demand matching problem (PDM) which combines aspects of the knapsack problem along with the b-matching problem. It is a generalization of the maximum weight matching problem which has been fundamental in the development of theory of computer science and operations research. This problem is NP-hard and there exists a constant ϵ>0 such that the problem admits no 1+ϵ-approximation algorithm, unless P=NP.Here, we investigate the performance of a distributed message passing algorithm called Max-sum belief propagation for computing the problem of finding the optimal perfect demand matching. As the main result, we demonstrate the rigorous theoretical analysis of the Max-sum BP algorithm for PDM, and establish that within pseudo-polynomial-time, our algorithm could converge to the optimal solution of PDM, provided that the optimal solution of its LP relaxation is unique and integral. Different from the techniques used in previous literature, our analysis is based on primal-dual complementary slackness conditions, and thus the number of iterations of the algorithm is independent of the structure of the given graph. Moreover, to the best of our knowledge, this is one of a very few instances where BP algorithm is proved correct for NP-hard problems.

Sub-6 GHz V2X-assisted MmWave optimal scheduling for vehicular networks

Sub-6 GHz assisted millimeter wave (mmWave) technology is a promising solution to address the ever-increasing data exchange demand in Vehicle-to-Everything (V2X) communication. The omni-directional sub-6 GHz channels exchange vehicle and control information in the vehicular network, while the directional mmWave technology is used for high-rate data transmission. Consider the huge amounts of vehicular data, the transmission constraints imposed by mmWave, and the unique vehicular ad hoc network (VANET) data features, a sub-6 GHz assisted-mmWave scheduling algorithm is needed to facilitate V2X data transmission. Existing scheduling algorithms in the literature fall short of addressing realistic V2X data features. Therefore, in this paper, we propose a novel objective to maximize the network utility that succinctly captures realistic V2X data features, including varying data sizes, data importance and their freshness decay with transmission delay. We formulate the scheduling problem into a novel mixed-integer sum-of-ratios optimization problem. To solve the challenging NP-complete problem, we first derive an equivalent parametric mixed-integer linear programming problem with existence and uniqueness proofs. Next, we provide the necessary condition for the optimality of the equivalent problem. A novel Parameterization-based Iterative Algorithm (PIA) is then developed to learn the global optimal solution with convergence analysis. Comparative simulation studies with the brute-force algorithm, the widely used branch and bound (BB) method, the greedy algorithm, and our previously developed maximum weight matching based iterative algorithm (MWMIA) are conducted to verify the correctness and effectiveness of the proposed PIA scheduling algorithm.

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

This paper considers a bipartite graph where a perfect matching does not necessarily exist. Linear programming is used in this paper as a special case of the linear program is the assignment problem, which is another name for the weighted maximum matching problem. The objective is to show that linear programming, in particular the simplex algorithm, can be used to calculate maximum weight matchings and minimum weighted vertex covers. This study also calculates and shows the equivalence of the maximum matching and minimum vertex cover cardinalities, and uses linear programming duality to present its relevance to König's theorem. Finally, Hall’s marriage theorem is used to explicitly prove the absence of a perfect matching in particular graphs. There exist many algorithms developed over the years that are designed to solve maximum matching problems in polynomial time, but the simplex method is one of the oldest and simplest algorithms to understand in solving these problems.

Joint resource scheduling and trajectory design for multi-UAV-assisted uplink NOMA networks

The combination of unmanned aerial vehicle (UAV) communication and non-orthogonal multiple access (NOMA) is of great significance to support massive connections in the Internet of things (IoT). This paper studies a multi-UAV-assisted communication network, where the UAVs are employed as aerial access points to receive data from IoT nodes with NOMA adopted for uplink transmission. The aim is to maximize the average sum rate of the network, with the requirements of the individual upload of each node. To handle the node clustering problem, the spectral clustering and maximum weight matching algorithm are first integrated to divide the IoT nodes. Then, the time-varying NOMA user grouping scheme is implemented accordingly. Furthermore, by considering the trajectory decomposition and successive hover-and-fly structure, the complex UAV trajectory problem is transformed into a linear programming and the optimal time-continuous trajectory is achieved. Extensive simulations validate the effectiveness of the proposed algorithms for the UAV-assisted NOMA network.

Automatic Parts Correspondence Determination for Transforming Assemblies via Local and Global Geometry Processing

Transforming assemblies are products that alter their shapes by re-assembling their parts. This idea is applied to a wide range of objects from folding gadgets as outdoor gear aimed at saving space, to robotic characters fighting in Hollywood films which drastically change their appearance. While the former type falls into a folding or packing problems, the latter requires a different viewpoint to be solved since the destination shape is not necessarily aiming at minimizing the occupation space. As a possible solution, this kind of deformation can be decomposed into segmentation of the shape to parts and parts matching. Segmentation is a general problem in shape modeling and numerous algorithms have been proposed for this. On the other hand, matching simultaneously multiple parts (many-to-many matching) has hardly been explored. This study develops a many-to-many matching algorithm for surface meshes of parts from two distinct destination shapes of a single transforming assembly. The proposed algorithm consists of a local geometry analysis and a global optimization of parts combination based on such analysis. For the local geometry analysis, the surface geometric feature is described by a local shape descriptor. Some vertices are detected as feature points by intrinsic shape signature (ISS) and the geometry at the feature points is expressed by the signature of histogram of orientation (SHOT). For all the combination of pairs from each destination shape, the number of feature points with similar descriptor values is counted. In the global optimization, the final matching is determined by the maximum weight matching on a complete bipartite graph whose nodes are the parts, and edges are weighted by the number of the feature points with similar descriptors. We present successful results for several examples to empirically show the effectiveness of the proposed algorithm.

Estimating optimal objective values for the TSP, VRP, and other combinatorial problems using randomization

AbstractApproximation of the optimal tour length in a Euclidean traveling salesman problem (TSP) has been studied by many researchers. In a previous study, we used the standard deviation in random tour lengths to approximate the optimal tour length in both Euclidean and non‐Euclidean TSPs and we obtained good estimates. In this paper, we show that the strong power‐law relationship between the standard deviation in random feasible solution values and the optimal solution value also holds for other Euclidean and near‐Euclidean combinatorial optimization problems like the minimum spanning tree (MST) and maximum weight matching (MWM) problems. We then enhance the estimation ability of the model by considering a second predictor: the mean in random feasible solution values. Experimental results show that by using the mean, standard deviation, and randomization, we can accurately predict the optimal solution values for the TSP, MST, MWM, and the capacitated vehicle routing problem (VRP).

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.

IRS-Assisted Physical Layer Security for 5G Enabled Industrial Internet of Things

5G is a key enabler of Industrial Internet of Things (IIoT) that provides seamless connectivity between machines, sensors and computing servers. Security and privacy are major concerns for 5G enabled IIoT. Physical Layer Security (PLS) is a promising technique that can enhance the security of 5G enabled IIoT. In this paper, we present an IRS-assisted PLS scheme for 5G enabled IIoT that improves the weighted secrecy sum-rate (WSSR) of the industrial wireless network, where eavesdroppers are present around the facility. The key idea is to jointly optimize active and passive beamforming vectors to increase the secrecy rate at the user. To maximize WSSR, we use the stable matching algorithm that optimally assigns IRSs for secure data sharing between industrial units. Simulation results show that the proposed scheme enhances the WSSR performance by 40% and minimum secrecy rate by 25% as compared to the random and maximum weight matching schemes, respectively.

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.

Sub-6 GHz V2X-Assisted Synchronous Millimeter Wave Scheduler for Vehicle-to-Vehicle Communications

As millimeter wave (mmWave) technology becomes more integrated into today's communication infrastructure, its application to mobile scenarios has attracted considerable interests. In particular, mmWave technology is one of the leading candidates in Vehicle-to-Everything (V2X) community to address the ever-rising data exchange demand from vehicles. However, the inherent directionality of the technology poses challenges to peer discovery and transmission scheduling in typical V2X scenarios, causing significant loss of network capacity. To mitigate this issue, we propose a novel sub-6 GHz V2X-assisted synchronous mmWave communication scheduler in this paper. The sub-6 GHz channel serves as control channel for mmWave scheduling. The scheduler carefully considers data importance and freshness in determining communication schedule, such that network utility across mmWave links is maximized. We formulate the scheduling problem into an integer linear programming (ILP) problem. However, ILP problem is NP-hard and has exponential computational complexity for the worst cases. We then develop a scheduling algorithm, maximum weight matching based iterative algorithm (MWMIA), that operates in polynomial time with performance guarantees for the NP-hard ILP, by leveraging structural properties of link conflicts. Comparative simulation studies verify the effectiveness of our scheduler against other solutions.

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.

Solving maximum weighted matching on large graphs with deep reinforcement learning

Maximum weighted matching (MWM), which finds a subset of vertex-disjoint edges with maximum weight, is a fundamental topic with a wide spectrum of applications in various fields, such as biotechnology, social analysis, web services, etc. However, traditional methods cannot achieve a good balance between cost and quality. Inspired by the reinforcement learning techniques, we propose an MWM framework, L2M, with a Deep Reinforcement Learning (DRL) model, which can efficiently and effectively solve the MWM problem on large-scale general graphs. First, in contrast to traditional DRL methods that define the Markov Decision Process (MDP) without considering the efficiency issue, we represent MWM as an MDP that is carefully designed to accelerate computation. In particular, this MDP supports selecting multiple edges for each action and uses a pruning method to reduce search space efficiently. Second, since none of the existing DRL methods can support edge selection on large-scale graphs efficiently, we propose an edge-message-passing network EEN to generate edge embedding. To the best of our knowledge, this is the first attempt that uses DRL model for solving the MWM problem. Experimental results show that L2M outperforms state-of-the-art algorithms and generalizes well on graphs of different sizes and distributions.

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.

Edge-Weighted Online Windowed Matching

Consider a matching problem, in which agents arrive to a marketplace over time and leave after some time periods. Agents can only be matched while present in the marketplace. Each pair of agents can yield a different match value, and a social planner seeks to maximize the total value from matches over a finite time horizon. First we study the case in which vertices arrive in an adversarial order. For the case when agents depart in the order of arrival, we provide a randomized [Formula: see text]-competitive algorithm. When departure times are drawn independently from a distribution with nondecreasing hazard rate, we establish a [Formula: see text]-competitive algorithm. When the arrival order is chosen uniformly at random and agents leave after a fixed number of time periods, a batching algorithm, which computes a maximum-weighted matching periodically, is shown to be 0.279-competitive.

Conserved Control Path in Multilayer Networks.

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that form multilayer networks. In this study, we describe a graph pattern called the conserved control path, which allows us to model a common control structure among different types of relations. We present a practical conserved control path detection method (CoPath), which is based on a maximum-weighted matching, to determine the paths that play the most consistent roles in controlling signal transmission in multilayer networks. As a pragmatic application, we demonstrate that the control paths detected in a multilayered pan-cancer network are statistically more consistent. Additionally, they lead to the effective identification of drug targets, thereby demonstrating their power in predicting key pathways that influence multiple cancers.

A Batch-dynamic Suitor Algorithm for Approximating Maximum Weighted Matching

Matching is a popular combinatorial optimization problem with numerous applications in both commercial and scientific fields. Computing optimal matchings w.r.t. cardinality or weight can be done in polynomial time; still, this task can become infeasible for very large networks. Thus, several approximation algorithms that trade solution quality for a faster running time have been proposed. For networks that change over time, fully dynamic algorithms that efficiently maintain an approximation of the optimal matching after a graph update have been introduced as well. However, no semi- or fully dynamic algorithm for (approximate) maximum weighted matching has been implemented. In this article, we focus on the problem of maintaining a \( 1/2 \) -approximation of a maximum weighted matching (MWM) in fully dynamic graphs. Limitations of existing algorithms for this problem are (i) high constant factors in their time complexity, (ii) the fact that none of them supports batch updates, and (iii) the lack of a practical implementation, meaning that their actual performance on real-world graphs has not been investigated. We propose and implement a new batch-dynamic \( 1/2 \) -approximation algorithm for MWM based on the Suitor algorithm and its local edge domination strategy [Manne and Halappanavar, IPDPS 2014]. We provide a detailed analysis of our algorithm and prove its approximation guarantee. Despite having a worst-case running time of \( \mathcal {O}(n + m) \) for a single graph update, our extensive experimental evaluation shows that our algorithm is much faster in practice. For example, compared to a static recomputation with sequential Suitor , single-edge updates are handled up to \( 10^5\times \) to \( 10^6\times \) faster, while batches of \( 10^4 \) edge updates are handled up to \( 10^2\times \) to \( 10^3\times \) faster.