Maximum Weight Matching Problem Research Articles

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.

Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a “prophet” who knows the future. An equally natural, though significantly less well-studied, benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? Motivated by applications in ride hailing, we study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of [Formula: see text]-competitive algorithms are known. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm, which approximates the optimal online algorithm within a factor of 0.51—strictly more than the best-possible constant against a prophet. In contrast, we show that it is PSPACE-hard to approximate this problem within some universal constant [Formula: see text]. Funding: Financial support from the National Science Foundation [Grants CCF1763970, CCF1812919, and CCF191070], the Office of Naval Research [Grant N000141912550], and Cisco Research is gratefully acknowledged.

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.

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.

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.

Resource Management for 5G NR Integrated Access and Backhaul: A Semi-Centralized Approach

The next generations of mobile networks will be deployed as ultra-dense networks, to match the demand for increased capacity and the challenges that communications in the higher portion of the spectrum (i.e., the mmWave band) introduce. Ultra-dense networks, however, require pervasive, high-capacity backhaul solutions, and deploying fiber optic to all base stations is generally considered to be too expensive for network operators. The 3rd Generation Partnership Project (3GPP) has thus introduced Integrated Access and Backhaul (IAB), a wireless backhaul solution in which the access and backhaul links share the same hardware, protocol stack, and also spectrum. The multiplexing of different links in the same frequency bands, however, introduces interference and capacity sharing issues, thus calling for the introduction of advanced scheduling and coordination schemes. This paper proposes a semi-centralized resource allocation scheme for IAB networks, designed to be flexible, with low complexity, and compliant with the 3GPP IAB specifications. We develop a version of the Maximum Weighted Matching (MWM) problem that can be applied on a spanning tree that represents the IAB network and whose complexity is linear in the number of IAB-nodes. The proposed solution is compared with state-of-the-art distributed approaches through end-to-end, full-stack system-level simulations with a 3GPP-compliant channel model, protocol stack, and a diverse set of user applications. The results show that our scheme can increase the throughput of cell-edge users up to 3 times, while decreasing the overall network congestion with an end-to-end delay reduction of up to 25 times.

A generalized ride-matching approach for sustainable shared mobility

On-demand shared mobility is a promising and sustainable transportation approach that can mitigate vehicle externalities, such as traffic congestion and emission. On-demand shared mobility systems require matching of one (one-to-one) or multiple riders (many-to-one) to a vehicle based on real-time information. We propose a novel Graph-based Many-to-One ride-Matching (GMOMatch) algorithm for the dynamic many-to-one matching problem in the presence of traffic congestion. GMOMatch, which is an iterative two-step method, provides high service quality and is efficient in terms of computational complexity. It starts with a one-to-one matching in Step 1 and is followed by solving a maximum weight matching problem in Step 2 to combine the travel requests. To evaluate the performance, it is compared with a ride-matching algorithm developed by Simonetto et al. (2019). Both algorithms are implemented in a micro-traffic simulator to assess their performance and their impact on traffic congestion in Downtown, Toronto road network. In comparison to the Simonetto, GMOMatch improved the service rate, vehicle kilometer traveled and traffic travel time by 32%, 16.07%, and 4%, respectively. The sensitivity analysis indicated that utilizing vehicles with a capacity of 10 can achieve 25% service rate improvement compared to a capacity of 4.

Binary Code Optimized for Partial Encryption

A common technique for the partial encryption of compressed images and videos encrypts only the sign bits of some syntax elements such as the quantized transform coefficients or the motion vector differences. The sign bit can be interpreted as the most significant bit (MSB) in the binary representation of the syntax element. Our work is motivated by the key observation that the binary code used for this representation has an impact on the quality of the reconstruction at the eavesdropper and on the size of the stream to be encrypted. Therefore, we address the problem of optimal binary code design for partial encryption. Ideally, the goal is to simultaneously maximize the eavesdropper’s distortion and minimize the length of the compressed MSB stream. Since these two objectives are conflicting in general, we formulate the problem as the maximization of a weighted sum of the eavesdropper’s distortion and of the probability of the MSB being 0. We cast the problem as a binary integer linear program equivalent to a maximum weight matching problem, which has a polynomial-time solution algorithm. We show that when the source to be quantized and the quantizer are symmetric, the problem can be converted to a linear program of a smaller size, for a family of distortion metrics. Extensive experiments assess the performance of the optimized binary code in comparison with existing approaches. The results reveal that certain existing partial encryption schemes could benefit from the proposed design.

AUTOMATED UAV LIDAR STRIP ALIGNMENT IN FORESTED AREAS USING DENSITY-BASED CANOPY CLUSTERING

Abstract. Recently, LiDAR point cloud data acquired by Unmanned Aerial Vehicles (UAVs) are used in many scientific disciplines and like the former photogrammetric techniques these data are usually collected in overlapping strips. Generation of comprehensive models of the scanned areas requires these strips to be aligned together which is a challenging process due to the multi sensor scanning system including the scanning sensor, the GNSS receiver and the IMU sensor. The main errors result from the inaccurate GNSS locations and flight path shifts as well as failure of the GNSS signals in complex urban or forest environments. For that reasons, the development of an automatic feature-dependant method in urban areas or individual tree-based in forest areas where there are no distinct features for strip adjustment in these environments become a must. This research work focuses on automated co-registration/alignment multiple point cloud strips of forested areas acquired from UAV LiDAR (or referred to as ULS) lack of artificial ground control. The main limitations of ULS data of forests are the relatively low sampling density of near ground areas and stem nullity due to the top-view scanning mode of ULS. To obviate this, this work explicates the tree crowns shape to identify the key points required for co-registration by applying a density based clustering algorithm (DBSCAN) to the tree crowns and models resulting clusters with Gaussian mixture models by learning the best parameters using maximum likelihood estimation to define the key points. A feature vector is assigned to each point by quantifying its angular and linear relationship with respect to the local system origin. Next, the similarity score matrix is computed by a fixed geometric relationship between the distance and angle similarity. Then, the maximum weight matching problem is solved for the similarity score to gain point-to-point correspondence. Finally, the optimal 2D rigid transformation parameters (one rotation and two translations without scale factor ) are obtained using permutations to try out for all possible paired combinations and count the number of inlier points satisfying a tolerance of planimetric deviation after alignment within a user defined threshold. The results of two test forest plots with different tree species and ULS point densities show a mean planimetric enhancement from 1.79 m to 0.22 ± 0.13 m for plot one and from 2.33 ± 0.53 m to 0.61 ± 0.21 m.

Graph Matching Via the Lens of Supermodularity

Graph matching, the problem of aligning a pair of graphs so as to minimize their edge disagreements, has received widespread attention owing to its broad spectrum of applications in data science. As the problem is NP–hard in the worst-case, a variety of approximation algorithms have been proposed for obtaining high quality, suboptimal solutions. In this article, we approach the task of designing an efficient polynomial-time approximation algorithm for graph matching from a previously unconsidered perspective. Our key result is that graph matching can be formulated as maximizing a monotone, supermodular set function subject to matroid intersection constraints. We leverage this fact to apply a discrete optimization variant of the minorization-maximization algorithm which exploits supermodularity of the objective function to iteratively construct and maximize a sequence of global lower bounds on the objective. At each step, we solve a maximum weight matching problem in a bipartite graph. Differing from prior approaches, the algorithm exploits the combinatorial structure inherent in the problem to generate a sequence of iterates featuring monotonically non-decreasing objective value while always adhering to the combinatorial matching constraints. Experiments on real-world data demonstrate the empirical effectiveness of the algorithm relative to the prevailing state-of-the-art.

A Distributed-Memory Algorithm for Computing a Heavy-Weight Perfect Matching on Bipartite Graphs

We design and implement an efficient parallel algorithm for finding a perfect matching in a weighted bipartite graph such that weights on the edges of the matching are large. This problem differs from the maximum weight matching problem, for which scalable approximation algorithms are known. It is primarily motivated by finding good pivots in scalable sparse direct solvers before factorization. Due to the lack of scalable alternatives, distributed solvers use sequential implementations of maximum weight perfect matching algorithms, such as those available in MC64. To overcome this limitation, we propose a fully parallel distributed memory algorithm that first generates a perfect matching and then iteratively improves the weight of the perfect matching by searching for weight-increasing cycles of length four in parallel. For most practical problems the weights of the perfect matchings generated by our algorithm are very close to the optimum. An efficient implementation of the algorithm scales up to 256 nodes (17,408 cores) on a Cray XC40 supercomputer and can solve instances that are too large to be handled by a single node using the sequential algorithm.

AMG based on compatible weighted matching for GPUs

We describe main issues and design principles of an efficient implementation, tailored to recent generations of Nvidia Graphics Processing Units (GPUs), of an Algebraic MultiGrid (AMG) preconditioner previously proposed by one of the authors and already available in the open-source package BootCMatch: Bootstrap algebraic multigrid based on Compatible weighted Matching for standard CPUs. The AMG method relies on a new approach for coarsening sparse symmetric positive definite (s.p.d.) matrices, named coarsening based on compatible weighted matching. It exploits maximum weight matching in the adjacency graph of the sparse matrix, driven by the principle of compatible relaxation, providing a suitable aggregation of unknowns which goes beyond the limits of the usual heuristics applied in the current methods. We adopt an approximate solution of the maximum weight matching problem, based on a recently proposed parallel algorithm, referred to as the Suitor algorithm, and show that it allows us to obtain good quality coarse matrices for our AMG on GPUs. We exploit inherent parallelism of modern GPUs in all the kernels involving sparse matrix computations both for the setup of the preconditioner and for its application in a Krylov solver, outperforming preconditioners available in the original sequential CPU code as well as the single node Nvidia AmgX library. Results for a large set of linear systems arising from discretization of scalar and vector partial differential equations (PDEs) are discussed.

A (2+ϵ)-Approximation for Maximum Weight Matching in the Semi-streaming Model

We present a simple deterministic single-pass (2+ϵ)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves on the currently best known approximation ratio of (4+ϵ). Our algorithm uses O ( n log 2 n ) bits of space for constant values of ϵ. It relies on a variation of the local-ratio theorem, which may be of use for other algorithms in the semi-streaming model as well.

Toward efficient and accurate function‐call graph matching of binary codes

SummaryReverse engineering, software plagiarism detection, and malware analysis have always been important issues in software and security fields. For a binary code, the function‐call graph (FCG) reflects its capability, structure, and intrinsic relations, which motivates us to study FCG matching and its applications in those problems systematically. In this work, we propose an FCG matching algorithm based on Hungarian algorithm that solves the maximum weight matching problem in polynomial time and makes matching between graphs of large scale possible. Also, optimizations including node pairs pruning and forward matching are proposed to improve the efficiency and accuracy of FCG matching algorithm. Finally, a series of experiments are conducted to show that FCG matching is an effective method and has huge application potentiality in software and security analysis.

Maximum Weight Matching Using Odd-Sized Cycles: Max-Product Belief Propagation and Half-Integrality

We study the maximum weight matching (MWM) problem for general graphs through the max-product belief propagation (BP) and related Linear Programming (LP). The BP approach provides distributed heuristics for finding the maximum a posteriori (MAP) assignment in a joint probability distribution represented by a graphical model (GM), and respective LPs can be considered as continuous relaxations of the discrete MAP problem. It was recently shown that a BP algorithm converges to the correct MAP/MWM assignment under a simple GM formulation of MWM, as long as the corresponding LP relaxation is tight. First, under the motivation for forcing the tightness condition, we consider a new GM formulation of MWM, say C-GM, using non-intersecting odd-sized cycles in the graph; the new corresponding LP relaxation, say C-LP, becomes tight for more MWM instances. However, the tightness of C-LP now does not guarantee such convergence and correctness of the new BP on C-GM. To address the issue, we introduce a novel graph transformation applied to C-GM, which results in another GM formulation of MWM, and prove that the respective BP on it converges to the correct MAP/MWM assignment, as long as C-LP is tight. Finally, we also show that C-LP always has half-integral solutions, which leads to an efficient BP-based MWM heuristic consisting of making sequential, “cutting plane”, modifications to the underlying GM. Our experiments show that this BP-based cutting plane heuristic performs, as well as that based on traditional LP solvers.

The Application of the Weighted k-Partite Graph Problem to the Multiple Alignment for Metabolic Pathways.

The problem of aligning multiple metabolic pathways is one of very challenging problems in computational biology. A metabolic pathway consists of three types of entities: reactions, compounds, and enzymes. Based on similarities between enzymes, Tohsato et al. gave an algorithm for aligning multiple metabolic pathways. However, the algorithm given by Tohsato et al. neglects the similarities among reactions, compounds, enzymes, and pathway topology. How to design algorithms for the alignment problem of multiple metabolic pathways based on the similarity of reactions, compounds, and enzymes? It is a difficult computational problem. In this article, we propose an algorithm for the problem of aligning multiple metabolic pathways based on the similarities among reactions, compounds, enzymes, and pathway topology. First, we compute a weight between each pair of like entities in different input pathways based on the entities' similarity score and topological structure using Ay et al.'s methods. We then construct a weighted k-partite graph for the reactions, compounds, and enzymes. We extract a mapping between these entities by solving the maximum-weighted k-partite matching problem by applying a novel heuristic algorithm. By analyzing the alignment results of multiple pathways in different organisms, we show that the alignments found by our algorithm correctly identify common subnetworks among multiple pathways.

New Algorithms for Maximum Weight Matching and a Decomposition Theorem

We revisit the classical maximum weight matching problem in general graphs with nonnegative integral edge weights. We present an algorithm that operates by decomposing the problem into W unweighted versions of the problem, where W is the largest edge weight. Our algorithm has running time as good as the current fastest algorithms for the maximum weight matching problem when W is small. One of the highlights of our algorithm is that it also produces an integral optimal dual solution; thus our algorithm also returns an integral certificate corresponding to the maximum weight matching that was computed. Our algorithm yields a new proof to the total dual integrality of Edmonds’ matching polytope and it also gives rise to a decomposition theorem for the maximum weight of a matching in terms of the maximum size of a matching in certain subgraphs. We also consider the maximum weight capacitated b-matching problem in bipartite graphs with nonnegative integral edge weights and show that it can also be decomposed into W unweighted versions of the problem, where W is the largest edge weight. Our second algorithm is competitive with known algorithms when W is small.

The Computational Complexity of Weighted Greedy Matching

Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDYMATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bushgraphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is ρ, then for every ε > 0 the randomized MRG algorithm of (Aronson et al. 1995) gives a (ρ − ε)-approximation for the maximum cardinality matching. We conjecture that a tightbound for ρ is 2/3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem.

Resource Allocation for Multiuser OFDMA Hybrid Full/Half-Duplex Relaying Systems With Direct Links

Resource allocation is an important strategy to optimize system performance in multiuser cooperative orthogonal frequency-division multiple-access (OFDMA) systems. In this paper, we jointly consider three different transmission modes in cooperative OFDMA systems, i.e., direct transmission (DT) mode, half-duplex (HD) relay cooperative transmission (HDRCT) mode, and full-duplex (FD) relay transmission (FDRT) mode. The joint optimization problem of transmission mode selection, subcarrier assignment, relay selection, subcarrier pairing, and power allocation (PA) is investigated. In this paper, we transform the binary assignment problem into a maximum-weighted bipartite matching problem, which can be solved by the classical Hungarian method. Based on the dual method, we solve the joint PA and binary assignment problem iteratively. Specifically, since the direct link is considered interference in the FD relay transmission mode, the PA problem in FD relay transmission mode is nontrivial. Thus, we provide a novel hierarchical dual method to solve the PA problem in FD relay transmission mode. In addition, in HDRCT mode, the joint transmission of both the source and relay is taken into account, and we provide a simple and insightful PA scheme. Simulation results show that the proposed algorithms can significantly enhance the overall system throughput, compared with previous works.

Constrained Resource Assignments: Fast Algorithms and Applications in Wireless Networks

Resource assignment problems occur in a vast variety of applications, from scheduling problems over image recognition to communication networks. Often these problems can be modeled by a maximum weight matching problem in (bipartite) graphs or generalizations thereof, and efficient and practical algorithms are known for these problems. Although in some of the applications an assignment of the resources may be needed only once, in many of these applications, the assignment has to be computed more often for different scenarios. In that case it is often essential that the assignments can be computed very fast. Moreover, implementing different assignments in different scenarios may come with a certain cost for the reconfiguration of the system. In this paper, we consider the problem of determining optimal assignments sequentially over a given time horizon, where consecutive assignments are coupled by constraints that control the cost of reconfiguration. We develop fast approximation and online algorithms for this problem with provable approximation guarantees and competitive ratios. Moreover, we present an extensive computational study about the applicability of our model and our algorithms in the context of orthogonal frequency division multiple access (OFDMA) wireless networks, finding a significant performance improvement for the total bandwidth of the system using our algorithms. For this application (the downlink of an OFDMA wireless cell) , the run time of matching algorithms is extremely important, having an acceptable range of a few milliseconds only. For the considered realistic instances, our algorithms perform extremely well: the solution quality is, on average, within a factor of 0.8–0.9 of optimal off-line solutions, and the running times are at most 5 ms per phase even in the worst case. Thus, our algorithms are well suited to be applied in the context of OFDMA systems. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2221 . This paper was accepted by Teck-Hua Ho, optimization.