Multidimensional Assignment Problem Research Articles

G-Aligner: a graph-based feature alignment method for untargeted LC–MS-based metabolomics

BackgroundLiquid chromatography–mass spectrometry is widely used in untargeted metabolomics for composition profiling. In multi-run analysis scenarios, features of each run are aligned into consensus features by feature alignment algorithms to observe the intensity variations across runs. However, most of the existing feature alignment methods focus more on accurate retention time correction, while underestimating the importance of feature matching. None of the existing methods can comprehensively consider feature correspondences among all runs and achieve optimal matching.ResultsTo comprehensively analyze feature correspondences among runs, we propose G-Aligner, a graph-based feature alignment method for untargeted LC–MS data. In the feature matching stage, G-Aligner treats features and potential correspondences as nodes and edges in a multipartite graph, considers the multi-run feature matching problem an unbalanced multidimensional assignment problem, and provides three combinatorial optimization algorithms to find optimal matching solutions. In comparison with the feature alignment methods in OpenMS, MZmine2 and XCMS on three public metabolomics benchmark datasets, G-Aligner achieved the best feature alignment performance on all the three datasets with up to 9.8% and 26.6% increase in accurately aligned features and analytes, and helped all comparison software obtain more accurate results on their self-extracted features by integrating G-Aligner to their analysis workflow. G-Aligner is open-source and freely available at https://github.com/CSi-Studio/G-Aligner under a permissive license. Benchmark datasets, manual annotation results, evaluation methods and results are available at https://doi.org/10.5281/zenodo.8313034ConclusionsIn this study, we proposed G-Aligner to improve feature matching accuracy for untargeted metabolomics LC–MS data. G-Aligner comprehensively considered potential feature correspondences between all runs, converting the feature matching problem as a multidimensional assignment problem (MAP). In evaluations on three public metabolomics benchmark datasets, G-Aligner achieved the highest alignment accuracy on manual annotated and popular software extracted features, proving the effectiveness and robustness of the algorithm.

Performance evaluation for multi-target tracking with temporal dimension specifics

With the great development of Multi-Target Tracking (MTT) technologies, many MTT algorithms have been proposed with their own advantages and disadvantages. Due to the fact that requirements to MTT algorithms vary from the application scenarios, performance evaluation is significant to select an appropriate MTT algorithm for the specific application scenario. In this paper, we propose a performance evaluation method on the sets of trajectories with temporal dimension specifics to compare the estimated trajectories with the true trajectories. The proposed method evaluates the estimate results of an MTT algorithm in terms of tracking accuracy, continuity and clarity. Furthermore, its computation is based on a multi-dimensional assignment problem, which is formulated as a computable form using linear programming. To enhance the influence of recent estimated states of the trajectories in the evaluation, an attention function is used to reweight the trajectory errors at different time steps. Finally, simulation results show that the proposed performance evaluation method is able to evaluate many aspects of the MTT algorithms. These evaluations are worthy for selecting suitable MTT algorithms in different application scenarios.

Centralized multi-sensor labeled multi-Bernoulli filter with partially overlapping fields of view

In this paper, the labeled multi-Bernoulli (LMB) filter is extended to the centralized multi-sensor (MS) case, resulting in a multi-sensor multi-object tracking (MS-MOT) algorithm having the ability to cope with partially overlapping fields-of-view (FoVs). Specifically, a joint prediction and update scheme for the MS-LMB filter is presented and the inherent NP-hard multi-dimensional assignment problem is solved via Gibbs sampling. We propose a suboptimal stationary distribution for the Gibbs sampling which admits conditionally independent sampling among multiple sensors thus dramatically decreasing the computational complexity. In addition, in order to tune object initialization when there is no background knowledge of the birth locations, a novel MS measurement-driven birth process is proposed. The proposed birth process has superior ability in terms of suppressing false tracks originating from clutter measurements of multiple sensors. The effectiveness of the proposed algorithm and birth process is demonstrated in simulation experiments concerning MS surveillance scenarios with partially overlapping sensor FoVs.

Landscape properties of the very large-scale and the variable neighborhood search metaheuristics for the multidimensional assignment problem

We study the recent metaheuristic search algorithm for the multidimensional assignment problem (MAP) using fitness landscape theory. The analyzed algorithm performs a very large-scale neighborhood search on a set of feasible solutions to the problem. We derive properties of the landscape graphs that represent these very large-scale search algorithms acting on the solutions of the MAP. In particular, we show that the search graph is a generalization of a hypercube. We extend and generalize the original very large-scale neighborhood search to develop the variable neighborhood search. The new search is capable of searching even larger large-scale neighborhoods. We perform numerical analyses of the search graph structures for various problem instances of the MAP and different neighborhood structures of the MAP algorithm based on a very large-scale search. We also investigate the correlation between fitness (i.e., objective values) and distance (i.e., path lengths) of the local minima (i.e., sinks of the landscape). Our results can be used to design improved search-based metaheuristics for the MAP.

Disambiguation of measurements for multiple acoustic source localization using deep multi-dimensional assignments

The location estimation of multiple simultaneously active sources in an acoustic sensor network is quite challenging because the correct combination of measurements received from different microphone arrays is not usually known, generally termed the data association problem. Existing techniques generally formulate the data association problem as a multidimensional assignment and solve them using classical optimal techniques. Although these approaches show effectiveness for a limited amount of active sources, the computational time and complexity of these techniques increase significantly when the microphones or acoustic sources increase. In this paper, a learning approach is proposed that solves the multidimensional assignment using a deep neural network. Initially, the features that are employed for the association are formulated for each of the detected sources in all the array nodes, and a multidimensional assignment problem is formulated. Subsequently, a deep multidimensional assignment network is devised to extract the correspondence probability of the measurements received from the microphone arrays. Specifically, a data-driven differentiable approach is presented for multidimensional assignments that is computationally efficient. The proposed methodology is validated under realistic conditions for both speech and urban signals. The methodology is compared to state-of-the-art methods for showing the performance gain in terms of accuracy of association and location estimation with a reduced computational time.

Multitarget Tracking Based on Dynamic Bayesian Network With Reparameterized Approximate Variational Inference

Multitarget tracking (MTT) is an important component of situation-awareness based on the Internet of Things (IoT). Existing algorithms mainly focus on tracking based on conventional measurements, e.g., bearings or ranges. However, measurement parameter estimations are considered in isolation, limiting the accuracy and resolution of MTT, and the related data association is an NP-hard multidimensional assignment problem. In this article, we develop a new one-step MTT algorithm based on a novel dynamic Bayesian network (DBN), i.e., DBNMTT. The new MTT algorithm directly infers target states from the raw measurement data by fusing the array signal model, the signal propagation model, and the motion model. In this new DBNMTT framework, we treat target states and conventional measurements, such as bearings and target energies as hidden random variables. The posterior joint probability optimization problem is translated into the problem of graphical model learning. In this way, we can improve the accuracy and resolution of MTT and convert the NP-hard data association problem to a hidden variable learning problem. For nonconjugate models in the DBNMTT, we develop a novel reparameterized approximation variational inference (ReAVI) approach to solve the learning problem. The ReAVI converts nonconjugate models to conjugate models with new parameters and reuses the mean-field algorithm. The performance of our proposed new MTT method, namely, DBNMTT based on ReAVI (DBNMTT-ReAVI), is analyzed on extensive simulations in challenging scenarios. The simulation results show that the DBNMTT-ReAVI algorithm is superior to conventional measurement-based MTT algorithms in several aspects, including the success probability, convergence, resolution, and accuracy.

Scalable Feature Matching Across Large Data Collections

This article is concerned with matching feature vectors in a one-to-one fashion across large collections of datasets. Formulating this task as a multidimensional assignment problem with decomposable costs (MDADC), we develop fast algorithms with time complexity roughly linear in the number n of datasets and space complexity a small fraction of the data size. These remarkable properties hinge on using the squared Euclidean distance as dissimilarity function, which can reduce matching problems between pairs of datasets to n problems and enable calculating assignment costs on the fly. To our knowledge, no other method applicable to the MDADC possesses these linear scaling and low-storage properties necessary to large-scale applications. In numerical experiments, the novel algorithms outperform competing methods and show excellent computational and optimization performances. An application of feature matching to a large neuroimaging database is presented. The algorithms of this article are implemented in the R package matchFeat available at github.com/ddegras/matchFeat. Supplementary materials for this article are available online.

Multidimensional Assignment Problem for Multipartite Entity Resolution

Multipartite entity resolution aims at integrating records from multiple datasets into one entity. We derive a mathematical formulation for a general class of record linkage problems in multipartite entity resolution across many datasets as a combinatorial optimization problem known as the multidimensional assignment problem. As a motivation for our approach, we illustrate the advantage of multipartite entity resolution over sequential bipartite matching. Because the optimization problem is NP-hard, we apply two heuristic procedures, a Greedy algorithm and very large scale neighborhood search, to solve the assignment problem and find the most likely matching of records from multiple datasets into a single entity. We evaluate and compare the performance of these algorithms and their modifications on synthetically generated data. We perform computational experiments to compare performance of recent heuristic, the very large-scale neighborhood search, with a Greedy algorithm, another heuristic for the MAP, as well as with two versions of genetic algorithm, a general metaheuristic. Importantly, we perform experiments to compare two alternative methods of re-starting the search for the former heuristic, specifically a random-sampling multi-start and a deterministic design-based multi-start. We find evidence that design-based multi-start can be more efficient as the size of databases grow large. In addition, we show that very large scale search, especially its multi-start version, outperforms simple Greedy heuristic. Hybridization of Greedy search with very large scale neighborhood search improves the performance. Using multi-start with as few as three additional runs of very large scale search offers some improvement in the performance of the very large scale search procedure. Last, we propose an approach to evaluating complexity of the very large-scale neighborhood search.

Multidimensional Assignment Problem in the medical sector: A case study

This paper studies a Multidimensional Assignment Problem (MAP) applied in the medical sector. The case study is taken from a major french medical service provider. This problem consists of assigning medical employees to perform healthcare or administrative tasks at workplaces within a predefined scheduling horizon. The system of constraints, such as ability, compatibility, availability, and accommodation capacity, characterizes the complexity of this study. The objective is to minimize the gap between the assigned workload and the target contractual workload. As far as we know, the Mixed-Integer Linear Programming (MILP) presented in this paper is the first model to handle such an objective. Numerical experiments are conducted on real-world instances to compare the performance of several well-known open-source and commercial solvers. The results serve as preliminary evidence for a feasibility study and allow us to select an appropriate approach for an efficient aid-decision system.

Market’s static model for software development based on a transport problem with non-fixed time additions

The authors describe the formulation of the market’s discrete-continuous static model for software development based on a transport problem with non-fixed time additions. In contrast to the existing transport problem with fixed cost surcharges, it is proposed to formulate a minimax setting in time with times that may contain a part that proportionate to the volumes of appointments. Thus, the authors come up with the idea of a hybrid formulation of transport problem with fixed cost surcharges and classical transport problem in time. Such problems arise when the total volume of vehicles which have to be used repeatedly on each route is limited, plus the fixed addition that arises taking into account the delay in making logistics decisions. It is shown that the set a discrete-continuous problem can be approximated from above by the classical transport problem in time, which can also be obtained according to the scheme used in the work by Balinski. The authors also describe an exact algorithm of the branch-and-bound method, based on the geometric interpretation of the problem, which decomposes into subproblems on non-empty faces of the polyhedral set of feasible solutions, which areconvex programming problems that can be numerically solved by the subgradient method described in work by Polyak. The calculation of the lower estimates of the criterion proposed in the work is reduced to the same problems. It is shown that the function of the best values of the criterion on the edges is not sub- or supermodular, as a function of a subset of pairs of indices corresponding to the positive values of the transportation volumes, which makes it impossible to use supermodular programming methods. In connection with the latter, the authors also describe the optimal polynomial version of the branch-and-bound method, obtained by analogy with the solution of the multidimensional assignment problem, and a numerical example of its use. The authors describe interpretation of transport problem with non-fixed additions as a generalized assignment problem with non-fixed price discounts, taking into account the difference between the wholesale and retail prices. The authors come up with the ideaof the application for building digital platforms in the software development market for loading tasks for performers.

Multi-Scan Multi-Sensor Multi-Object State Estimation

If computational tractability were not an issue, multi-object estimation should integrate all measurements from multiple sensors across multiple scans. In this article, we propose an efficient numerical solution to the multi-scan multi-sensor multi-object estimation problem by computing the (labeled) multi-sensor multi-object posterior density. Minimizing the $L_{1}$-norm error from the exact posterior density requires solving large-scale multi-dimensional assignment problems that are NP-hard. An efficient multi-dimensional assignment algorithm is developed based on Gibbs sampling, together with convergence analysis. The resulting multi-scan multi-sensor multi-object estimation algorithm can be applied either offline in one batch or recursively. The efficacy of the algorithm is demonstrated using numerical experiments with a simulated dataset.

Thresholds versus fractional expectation-thresholds

Proving a conjecture of Talagrand, a fractional version of the ``expectation-threshold" conjecture of Kalai and the second author, we show that $p_c (\mathcal{F}) = O(q_f(\mathcal{F})\mathrm{log}\ \ell(\mathcal{F}))$ for any increasing family $\mathcal{F}$ on a finite set $X$, where $p_c(\mathcal{F})$ and $q_f(\mathcal{F})$ are the threshold and ``fractional expectation-threshold" of $\mathcal{F}$, and $\ell(\mathcal{F})$ is the maximum size of a minimal member of $\mathcal{F}$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson--Kahn--Vu), bounded degree spanning trees (Montgomery), and bounded degree graphs (new). We also resolve (and vastly extend) the ``axial" version of the random multi-dimensional assignment problem (earlier considered by Martin--Mézard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős--Rado ``Sunflower Conjecture."

Fusion-Based Multidetection Multitarget Tracking With Random Finite Sets

Multidetection (MD) systems are characterized by multiple observation modes (OMs), and hence, simultaneously produce multiple measurements for each target. The key challenge in exploiting MD systems for multitarget tracking (MTT), compared to single-detection (SD) systems, is the significant amount of extra computational burden involved in order to solve the resulting multidimensional assignment problem among measurements, targets, and OMs. This article presents a novel computationally efficient MTT framework for MD systems, wherein the multitarget state is modeled as a random finite set (RFS), and a bank of OM-dependent MTT RFS filters with SD model are employed to recursively provide OM-dependent posteriors. The latter, which contain both real and false targets, are then suitably fused so as to enhance consensus on the true targets while weakening trust on the existence of the false ones. In this way, the computational complexity is significantly reduced compared to existing MTT algorithms with the MD model. Two representative RFS filters, i.e., unlabeled probability hypothesis density (PHD) and labeled multi-Bernoulli (LMB), are considered in the proposed framework and the computational complexity of the resulting MD MTT algorithms is analyzed. Performance of the proposed approach is assessed by simulation experiments in both over-the-horizon-radar (OTHR) and single-frequency-network passive radar (SFN-PR) MTT applications.

Developing an efficient optimization of course-lecturer distribution timetabling using transportation algorithm: A case study of Federal Polytechnic Offa

The institution course timetabling problem (ICTP) is a multidimensional assignment-problem that varies from course timetabling, class-teacher timetabling, student scheduling, teacher assignment, and classroom assignment. Many researchers have attempted to solve problems as related to timeslot but neglecting areas of course allocation to lecturers. The paper presented a course allocation and distribution model for lecturers based on their fields of interest and qualification to a transportation algorithm which was aimed at optimising the performance of lecturers in each course. It also evaluated overall efficiency of lecturers without exceeding the maximum workload. The performance of the course-to-lecturer allocation of the electrical/electronic engineering department, federal Polytechnic Offa, Kwara State was collected using simple questionnaire. The information obtained from the questionnaire was used to test the Algorithm developed. The result showed that using the developed algorithm for course distribution, the performance is 76.98% and 82.1% for the first and second semesters respectively. This showed that using the algorithm for allocation of courses to the lecturers of any department can be done based on input data without exceeding the recommended workloads of each cadre. This improved the quality of teaching, save time, and resources compare with manual methods. The study therefore recommended that future work should include practical distribution among technologists, sharing the excess workload to a particular lecturing grade as the case may be.

Group Target Tracking Based on MS-MeMBer Filters

This paper presents a new group target tracking method based on the standard multi-sensor multi-target multi-Bernoulli (MS-MeMBer) filter. In the prediction step, the group structure is used to constrain the movement of the constituent members within the respective groups. Specifically, the group of members is considered as an undirected random graph. Combined with the virtual leader-follower model, the motion equation of the members within groups is formulated. In the update step, the partitioning problem of multiple sensors is transformed into a multi-dimensional assignment (MDA) problem. Compared with the original two-step greedy partitioning mechanism, the MDA algorithm achieves better measurement partitions in group target tracking scenarios. To evaluate the performance of the proposed method, a simulation scenario including group splitting and merging is established. Results show that, compared with the standard MS-MeMBer filter, our method can effectively estimate the cardinality of members and groups at the cost of increasing computational load. The filtering accuracy of the proposed method outperforms that of the MS-MeMBer filter.

A dual approach to multi-dimensional assignment problems

In this paper, we extend the purely dual formulation that we recently proposed for the three-dimensional assignment problems to solve the more general multidimensional assignment problem. The convex dual representation is derived and its relationship to the Lagrangian relaxation method that is usually used to solve multidimensional assignment problems is investigated. Also, we discuss the condition under which the duality gap is zero. It is also pointed out that the process of Lagrangian relaxation is essentially equivalent to one of relaxing the binary constraint condition, thus necessitating the auction search operation to recover the binary constraint. Furthermore, a numerical algorithm based on the dual formulation along with a local search strategy is presented. The simulation results show that the proposed algorithm outperforms the traditional Lagrangian relaxation approach in terms of both accuracy and computational efficiency.

Machine Learning Methods for Data Association in Multi-Object Tracking

Data association is a key step within the multi-object tracking pipeline that is notoriously challenging due to its combinatorial nature. A popular and general way to formulate data association is as the NP-hard multi-dimensional assignment problem. Over the past few years, data-driven approaches to assignment have become increasingly prevalent as these techniques have started to mature. We focus this survey solely on learning algorithms for the assignment step of multi-object tracking, and we attempt to unify various methods by highlighting their connections to linear assignment and to the multi-dimensional assignment problem. First, we review probabilistic and end-to-end optimization approaches to data association, followed by methods that learn association affinities from data. We then compare the performance of the methods presented in this survey and conclude by discussing future research directions.

On random multi-dimensional assignment problems

We study random multidimensional assignment problems where the costs decompose into the sum of independent random variables. In particular, in three dimensions, we assume that the costs Wi,j,k satisfy Wi,j,k=ai,j+bi,k+cj,k where the ai,j,bi,k,cj,k are independent uniform [0,1] random variables. Our objective is to minimise the total cost and we show that w.h.p. a simple greedy algorithm is a (3+o(1))-approximation. This is in contrast to the case where the Wi,j,k are independent exponential rate 1 random variables. Here all that is known is an no(1)-approximation, due to Frieze and Sorkin.

GPU-accelerated Lagrangian heuristic for multidimensional assignment problems with decomposable costs

In this paper, we describe a GPU-accelerated parallel algorithm for the axial Multidimensional Assignment Problem with Decomposable Costs (MDADC), which is one of the most fundamental formulations for data association. MDADC is known to be NP-hard and is large-dimensioned in most realistic cases; hence, heuristic solutions with qualified optimality gaps is the best one can hope for, given the state-of-knowledge. The main contribution of this paper is an efficient parallelization of the Lagrangian subgradient search algorithm specifically targeted towards the Graphics Processing Units (GPUs) based on the NVIDIA Compute Unified Device Architecture (CUDA). A GPU-accelerated Linear Assignment Problem (LAP) solver is leveraged in concert with the Lagrangian scheme for further speed-up. We also implemented a multi-GPU variant of this algorithm which maintains a good speedup profile, when tested on problems with 31 billion variables, on up to 128 GPUs.

A Metric on the Space of Finite Sets of Trajectories for Evaluation of Multi-Target Tracking Algorithms

In this paper, we propose a metric on the space of finite sets of trajectories for assessing multi-target tracking algorithms in a mathematically sound way. The main use of the metric is to compare estimates of trajectories from different algorithms with the ground truth of trajectories. The proposed metric includes intuitive costs associated to localization error for properly detected targets, missed and false targets and track switches at each time step. The metric computation is based on solving a multi-dimensional assignment problem. We also propose a lower bound for the metric, which is also a metric for sets of trajectories and is computable in polynomial time using linear programming. We also extend the proposed metrics on sets of trajectories to random finite sets of trajectories.