Bipartite Matching Problem Research Articles

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most rn from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for G(n,rn) in the thermodynamic limit with nrnd= const., and also in the dense limit with nrnd→∞, rn→0. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the traveling salesman, spanning tree, matching, bipartite matching and bipartite traveling salesman problems, for a general class of weight functions with at most polynomial growth of order d−ε, under thermodynamic scaling of the distance parameter.

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

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.

Packing returning secretaries

AbstractWe study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to determine a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5‐competitive algorithm. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 − o(1), and an algorithm with ratio at least 0.5459 for all n ≥ 1. We extend all algorithms and ratios to k ≥ 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed. We focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Θ(n log n) is always sufficient. For bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(r′ log(n/r′)), where r′ is the minimum rank of the matroid and the dual matroid.

Linear Matroid Intersection is in Quasi-NC

Given two matroids on the same ground set, the matroid intersection problem asks to find a common independent set of maximum size. In case of linear matroids, the problem had a randomized parallel algorithm but no deterministic one. We give an almost complete derandomization of this algorithm, which implies that the linear matroid intersection problem is in quasi-NC. That is, it has uniform circuits of quasi-polynomial size $$n^{O(\log n)}$$ and O(polylog(n)) depth. Moreover, the depth of the circuit can be reduced to O(log2 n) in case of zero characteristic fields. This generalizes a similar result for the bipartite perfect matching problem. Our main technical contribution is to derandomize the Isolation lemma for the family of common bases of two matroids. We use our isolation result to give a quasi-polynomial time blackbox algorithm for a special case of Edmonds' problem, i.e., singularity testing of a symbolic matrix, when the given matrix is of the form $$A_{0} + A_{1 }x_{1} + \cdots + A_{m} x_{m}$$ , for an arbitrary matrix A0 and rank-1 matrices $$A_{1}, A_{2}, \dots, A_{m}$$ . This can also be viewed as a blackbox polynomial identity testing algorithm for the corresponding determinant polynomial. Another consequence of this result is a deterministic solution to the maximum rank matrix completion problem. Finally, we use our result to find a deterministic representation for the union of linear matroids in quasi-NC.

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.

Exploiting relay nodes for maximizing wireless underground sensor network lifetime

A major challenge in wireless underground sensor networks is the signal attenuation originated from multi-environment transmission between underground sensor nodes and the above-ground base station. To overcome this issue, an efficient approach is deploying a set of relay nodes aboveground, thereby reducing transmission loss by shortening transmitting distance. However, this introduces several new challenges, including load balancing and transmission loss minimization. This paper tackles the problem of deploying relay nodes to reduce transmission loss under a load balancing constraint by proposing two approximation algorithms. The first algorithm is inspired by Beam Search, combined with a new selection scheme based on Boltzmann distribution. The second algorithm aims to further improve the solutions obtained by the former by reducing the transmission loss. We observe that we can find an optimal assignment between sensor nodes and a set of the chosen relay in polynomial time by reformulating the part of the problem as a bipartite matching problem with minimum cost. Experimental results indicate that the proposed methods perform better than the other existing ones in most of our test instances while reducing the execution time.

Maximum Matching in the Online Batch-arrival Model

Consider a two-stage matching problem, where edges of an input graph are revealed in two stages (batches) and in each stage we have to immediately and irrevocably extend our matching using the edges from that stage. The natural greedy algorithm is half competitive. Even though there is a huge literature on online matching in adversarial vertex arrival model , no positive results were previously known in adversarial edge arrival model . For two-stage bipartite matching problem, we show that the optimal competitive ratio is exactly 2/3 in both the fractional and the randomized-integral models. Furthermore, our algorithm for fractional bipartite matching is instance optimal , i.e., it achieves the best competitive ratio for any given first stage graph. We also study natural extensions of this problem to general graphs and to s stages and present randomized-integral algorithms with competitive ratio ½ + 2− O(s) . Our algorithms use a novel Instance-Optimal-LP and combine graph decomposition techniques with online primal-dual analysis.

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.

Quantum Optimal Transport is Cheaper

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.

On random stable matchings: Cyclic ones with strict preferences and two-sided ones with partially ordered preferences

Consider a cyclically ordered collection of r equi-numerous “agent” sets with strict preferences of every agent over the agents from the next set. A weakly stable cyclic matching is a partition of the set of agents into disjoint union of r-long cycles, one agent from each set per cycle, such that there are no destabilizing r-long cycles, i.e. cycles in which every agent strictly prefers its successor to its successor in the matching. Assuming that the preferences are uniformly random and independent, we show that the expected number of stable matchings grows with n (cardinality of each agent set) as (nlog⁡n)r−1. Next we consider a bipartite stable matching problem where preference list of each agent forms a partially ordered set. Each partial order is an intersection of several, ki for side i, independent, uniformly random, strict orders. For k1+k2>2, the expected number of stable matchings is analyzed for three, progressively stronger, notions of stability. The expected number of weakly stable matchings is shown to grow super-exponentially fast. In contrast, for min⁡(k1,k2)>1, the fraction of instances with at least one strongly stable (super-stable) matching is super-exponentially small.

Online advertising assignment problem without free disposal

This paper presents an online advertising assignment problem that generalizes the online version of the bipartite matching problem. Specifically, it focuses on the Display Ads problem, which is a generalization of the edge-weighted and capacitated matching problem. The display ads problem has been studied alongside the property of free disposal, in which an advertisement is allowed to be matched more times than its capacity. Although the problem with free disposal is tractable, the problem situation might be restricted and challenging to apply to other types of problems. The objective of this research on the display ads problem is to maximize the total weight of matched edges while considering a strict capacity constraint. This paper analyzes two online input orders (adversarial and probabilistic orders) to the problem. For the adversarial order, we design deterministic algorithms with worst-case guarantees and prove the competitive ratios of them. Upper bounds for the problem are also proposed. For the probabilistic order, stochastic online algorithms, consisting of scenario-based stochastic programming and Benders decomposition, are presented. We conduct numerical experiments of the stochastic online algorithm in two probabilistic order models (known IID and random permutation).

Joint Relay Assignment and Power Allocation for Multiuser Multirelay Networks Over Underwater Wireless Optical Channels

Multiuser multirelay network is a potential scenario to fulfill the transmission requirements of various sources and high-volume traffic for the Internet of Underwater Things. To efficiently complete concurrent transmissions for multiple users, this article investigates the joint relay assignment and power allocation problem for multiuser multirelay networks based on the underwater optical wireless communication (UOWC). Specifically, the multiuser multirelay network for UOWC based on decode-and-forward relaying is modeled, where the absorption, scattering, solar radiation noise, and oceanic turbulence of UOWC are all considered. The joint optimization problem of relay assignment and power allocation is formulated as a mixed-integer programming problem, where the average outage probability is minimized with the constraint of total transmitted power. To solve this joint problem, an alternating optimization method is employed, which alternately optimizes the relay assignment and power allocation subproblems. The relay assignment subproblem is modeled as a weighted bipartite matching problem and solved by an improved Kuhn–Munkres algorithm, whereas the power allocation subproblem is proved to be quasiconvex and solved by an iterative bisection algorithm. The simulation results indicate that the proposed schemes significantly reduce the average outage probability with fast convergence.

Privacy preserving online matching on ridesharing platforms

Ridesharing platforms, as typical applications of spatial crowdsourcing, are becoming more and more popular in the era of mobile internet and sharing economy. One of the most fundamental issues on ridesharing platforms is to assign orders to drivers, which can be naturally modeled as online bipartite matching problem. However, conventional online matching algorithms usually lack data privacy protection mechanisms. This has become a serious issue since the spatiotemporal data of passengers is often sensitive. New policies such as EU’s General Data Protection Regulation (GDPR) also enforce protection of sensitive data, which further exacerbate the privacy issues. To deal with the problems, in this paper we propose a framework based on differential privacy (DP) techniques to preserve the privacy of individuals on ridesharing platforms. Specifically, we devise a novel approach to perturb locations in online minimum bipartite matching problem and theoretically show that the performance of the perturbed matching algorithm has the same magnitude with the unperturbed one. Experiments conducted on real datasets have also shown the effectiveness of proposed framework.

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.

Finding strongly popular [formula omitted]-matchings in bipartite graphs

The computational complexity of the bipartite popular matching problem depends on how indifference appears in the preference lists. If one side has strict preferences while nodes on the other side are all indifferent (but prefer to be matched), then a popular matching can be found in polynomial time (Cseh et al., 2015). However, as the same paper points out, the problem becomes NP-complete if nodes with strict preferences are allowed on both sides and indifferent nodes are allowed on one side. We show that the problem of finding a strongly popular matching is polynomial-time solvable even in the latter case. More generally, we give a polynomial-time algorithm for the many-to-many version, i.e. the strongly popular b-matching problem, in the setting where one side has strict preferences while agents on the other side may have one tie of arbitrary length at the end of their preference list.

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.

Pricing Carpool Rides Based on Schedule Displacement

This paper considers a carpool matching (CaMa) problem in which participants price shared rides based on both operating cost and schedule displacement (i.e., the absolute difference between the desired and actual arrival times). By reporting their valuation of this displacement, each participant in effect bids for every possible shared ride that generates a unique value to her. The CaMa problem can be formulated as a mixed integer program (MIP) that maximizes the social welfare by choosing matching pairs and a departure time for each pair. We show the optimal departure time can be determined for each pair a priori, independent of the matching problem. This result reduces the CaMa problem to a standard bipartite matching problem. We prove that the classical Vickrey-Clarke-Groves (VCG) pricing policy ensures no participant is worse off or has the incentive to misreport their valuation of schedule displacement. To control the large deficit created by the VCG policy, we develop a single-side reward (SSR) pricing policy, which only compensates participants who are forced by the system to endure a schedule displacement. Under the assumption of overpricing tendency (i.e., no participant would want to underreport their value), we show the SSR policy not only generates substantial profits, but also retains the other desired properties of the VCG policy, notably truthful reporting. Even though it cannot rule out underreporting, our simulation experiments confirm that the SSR policy is a robust and deficit-free alternative to the VCG policy. Specifically, we find that (1) underreporting is not a practical concern for a carpool platform as it never reduces the number of matched pairs and its impact on profits is largely negligible; and (2) participants have very little to gain by underreporting their value.

Dynamic dispatching and repositioning policies for fast-response service networks

We address the problem of dispatching and pro-actively repositioning service resources in service networks such that fast responses to service requests are realized in a cost-efficient way. By formulating this problem as a Markov decision process, we are able to investigate the structure of the optimal policy in the application domain of service logistics. Using these insights, we then propose scalable dynamic heuristics for both the dispatching and repositioning sub-problem, based on the minimum weighted bipartite matching problem and the maximum expected covering location problem, respectively. The dynamic dispatching heuristic takes into account real-time information about both the state of equipment and the fleet of service engineers, while the dynamic repositioning heuristic maximizes the expected weighted coverage of future service requests. In a test bed with a small network, we show that our most advanced heuristic performs well with an average optimality gap of 4.3% for symmetric instances and 5.8% for asymmetric instances. To show the practical value of our proposed heuristics, extensive numerical experiments are conducted on a large test bed with service logistics networks of real-life size where significant savings of up to 56% compared to a state-of-the-art benchmark policy are attained.

Pricing Carpool Rides Based on Schedule Displacement

This paper considers a carpool matching (CaMa) problem in which participants price shared rides based on both operating cost and schedule displacement (i.e, the absolute difference between the desired and actual arrival times). By reporting their valuation of this displacement, each participant in effect bids for every possible shared ride that generates a unique value to her. The CaMa problem can be formulated as a mixed integer program (MIP) that maximizes the social welfare by choosing matching pairs and a departure time for each pair. We show the optimal departure time can be determined for each pair a priori, independent of the matching problem. This result reduces the CaMa problem to a standard bipartite matching problem. We prove that the classical Vickrey-Clarke-Groves (VCG) pricing policy ensures no participant is worse off or has the incentive to misreport their valuation of schedule displacement. To control the large deficit created by the VCG policy, we develop a single-side reward (SSR) pricing policy, which only compensates participants who are forced by the system to endure a schedule displacement. Under the assumption of overpricing tendency (i.e., no participant would want to underreport their value), we show the SSR policy not only generates substantial profits, but also retains the other desired properties of the VCG policy, notably truthful reporting. Even though it cannot rule out underreporting, our simulation experiments confirm that the SSR policy is a robust and deficit-free alternative to the VCG policy. Specifically, we find that (1) underreporting is not a practical concern for a carpool platform as it never reduces the number of matched pairs and its impact on profits is largely negligible; and (2) participants have very little to gain by underreporting their value.