The anticlustering problem is to partition a set of objects into K equal-sized anticlusters such that the sum of distances within anticlusters is maximized. The anticlustering problem is NP-hard. We focus on anticlustering in Euclidean spaces, where the input data is tabular and each object is represented as a D-dimensional feature vector. Distances are measured as squared Euclidean distances between the respective vectors. Applications of Euclidean anticlustering include social studies, particularly in psychology, K-fold cross-validation in which each fold should be a good representative of the entire dataset, the creation of mini-batches for gradient descent in neural network training, and balanced K-cut partitioning. In particular, machine-learning applications involve million-scale datasets and very large values of K, making scalable anticlustering algorithms essential. Existing algorithms are either exact methods that can solve only small instances or heuristic methods, among which the most scalable is the exchange-based heuristic fast_anticlustering. We propose a new algorithm, the Assignment-Based Anticlustering algorithm (ABA), which scales to very large instances. A computational study shows that ABA outperforms fast_anticlustering in both solution quality and running time. Moreover, ABA scales to instances with millions of objects and hundreds of thousands of anticlusters within short running times, beyond what fast_anticlustering can handle. As a balanced K-cut partitioning method for tabular data, ABA is superior to the well-known METIS method in both solution quality and running time. The code of the ABA algorithm is available on GitHub.

Clustering is an unsupervised machine learning task that consists of identifying groups of similar objects. It has numerous applications and is increasingly used in fairness-sensitive domains where objects represent individuals, such as customers, employees, or students. We address a fair clustering problem in which objects belong to protected groups. The problem consists of partitioning the objects into a predefined number of clusters while attaining a user-defined target level of fairness, meaning that each protected group is sufficiently represented in each cluster. The objective is to minimize the clustering cost, defined as the sum of squared Euclidean distances between the objects and the centers of their clusters. Since clustering cost and fairness are generally in conflict, managing the trade-off between them is essential in practical applications. Existing methods provide limited control over this trade-off and either fail to scale to large datasets or, when they scale, produce low-quality solutions. We propose a general framework for fair clustering that provides precise control over the cost-fairness trade-off and introduce three heuristics based on it. The first heuristic focuses on solution quality and the flexibility to incorporate additional constraints, the second improves scalability while retaining high solution quality, and the third is designed for maximum scalability, producing solutions for instances with millions of objects in seconds. The proposed heuristics outperform existing approaches in comprehensive numerical experiments on benchmark datasets. The source code of our heuristics and instructions for reproducing the experiments are publicly available on GitHub.

The Quadratic Knapsack Problem (QKP) involves selecting a subset of elements that maximizes the sum of pairwise and singleton utilities without exceeding a given budget. The pairwise utilities are nonnegative, the singleton utilities may be positive, negative, or zero, and the node costs are nonnegative. We introduce a Breakpoints Algorithm for QKP, named QKBP, which is based on a technique proposed in Hochbaum (2009) for efficiently generating the concave envelope of the solutions to the relaxation of the problem for all values of the budget. Our approach utilizes the fact that breakpoints in the concave envelopes are optimal solutions for their respective budgets. For budgets between breakpoints, a fast greedy heuristic derives high-quality solutions from the optimal solutions of adjacent breakpoints. The QKBP algorithm is a heuristic which is highly scalable due to an efficient parametric cut procedure used to generate the concave envelope. This efficiency is further improved by a newly developed compact problem formulation. Our extensive computational study on both existing and new benchmark instances, with up to 10,000 elements, shows that while some leading algorithms perform well on a few instances, QKBP consistently delivers high-quality solutions regardless of instance size, density, or budget. Moreover, QKBP achieves these results in significantly faster running times than all leading algorithms. The source code of the QKBP algorithm, the benchmark instances, and the detailed results are publicly available on GitHub.

We study here the semi-supervised k-clustering problem where information is available on whether pairs of objects are in the same or in different clusters. This information is either available with certainty or with a limited level of confidence. We introduce the PCCC (Pairwise-Confidence-Constraints-Clustering) algorithm, which iteratively assigns objects to clusters while accounting for the information provided on the pairs of objects. Our algorithm uses integer programming for the assignment of objects which allows to include relationships as hard constraints that are guaranteed to be satisfied or as soft constraints that can be violated subject to a penalty. This flexibility distinguishes our algorithm from the state-of-the-art in which all pairwise constraints are either considered hard, or all are considered soft. We developed an enhanced multi-start approach and a model-size reduction technique for the integer program that contributes to the effectiveness and the efficiency of the algorithm. Unlike existing algorithms, our algorithm scales to large-scale instances with up to 60,000 objects, 100 clusters, and millions of cannot-link constraints (which are the most challenging constraints to incorporate). We compare the PCCC algorithm with state-of-the-art approaches in an extensive computational study. Even though the PCCC algorithm is more general than the state-of-the-art approaches in its applicability, it outperforms the state-of-the-art approaches on instances with all hard or all soft constraints both in terms of runtime and various metrics of solution quality. The code of the PCCC algorithm is publicly available on GitHub.

The PCCC algorithm is a clustering method that incorporates both hard and soft must-link and cannot-link constraints.

Clustering is the process of grouping similar objects based on their features. In many real-world clustering applications where the objects refer to persons, there is a great need to ensure that the resulting clusters are fair and unbiased. Such applications have led to the emergence of novel types of clustering problems. We consider here the fair-capacitated clustering problem which consists of partitioning a set of objects into a predefined number of clusters subject to fairness and cardinality constraints. The state-of-the-art algorithm for this problem considers the fairness and the cardinality constraints in two separate steps. We introduce here a new model-based approach that considers the two types of constraints simultaneously. In a computational comparison based on benchmark instances from the literature, we demonstrate that our algorithm finds substantially better solutions than the state-of-the-art algorithm in similar running time.

In direct marketing, companies use sales campaigns to target their customers with personalized product offers. The effectiveness of direct marketing greatly depends on the assignment of customers to campaigns. In this paper, we consider a real-world planning problem of a major telecommunications company that assigns its customers to individual activities of its direct marketing campaigns. Various side constraints, such as budgets and sales targets, must be met. Conflict constraints ensure that individual customers are not assigned too frequently to similar activities. Related problems have been addressed in the literature; however, none of the existing approaches cover all the side constraints considered here. To close this gap, we develop a matheuristic that employs a new decomposition strategy to cope with the large number of conflict constraints in typical problem instances. In a computational experiment, we compare the performance of the proposed matheuristic to the performance of two mixed-binary linear programs on a test set that includes large-scale real-world instances. The matheuristic derives near-optimal solutions in short running times for small- to medium-sized instances and scales to instances of practical size comprising millions of customers and hundreds of activities. The deployment of the matheuristic at the company has considerably increased the overall effectiveness of its direct marketing campaigns. (C) 2022 The Author(s). Published by Elsevier B.V.

We consider a semi-supervised $k$-clustering problem where information is available on whether pairs of objects are in the same or in different clusters. This information is either available with certainty or with a limited level of confidence. We introduce the PCCC algorithm, which iteratively assigns objects to clusters while accounting for the information provided on the pairs of objects. Our algorithm can include relationships as hard constraints that are guaranteed to be satisfied or as soft constraints that can be violated subject to a penalty. This flexibility distinguishes our algorithm from the state-of-the-art in which all pairwise constraints are either considered hard, or all are considered soft. Unlike existing algorithms, our algorithm scales to large-scale instances with up to 60,000 objects, 100 clusters, and millions of cannot-link constraints (which are the most challenging constraints to incorporate). We compare the PCCC algorithm with state-of-the-art approaches in an extensive computational study. Even though the PCCC algorithm is more general than the state-of-the-art approaches in its applicability, it outperforms the state-of-the-art approaches on instances with all hard constraints or all soft constraints both in terms of running time and various metrics of solution quality. The source code of the PCCC algorithm is publicly available on GitHub.

The design of supply networks that are resilient to disruptions has recently attracted considerable attention. We consider supply networks where a set of clients are served from a set of facilities. The cost of serving a client from a facility is proportional to the distance between the client and the facility. When a facility becomes unavailable due to a disruption, its clients are reassigned to the closest facility that is still operating. The network is resilient when disruptions cause only moderate reassignment costs. One way to design a resilient network is to solve the fault-tolerant k-median problem. Under this problem, a set of k facilities (medians) must be located such that the sum of distances from clients to their r nearest facilities is minimized. This paper introduces a new algorithm for large-scale instances of this problem. Using a benchmark instance with close to 10,000 clients, we demonstrate that our heuristic consistently devises better solutions than the state-of-the-art approach in much shorter running times.

Reducing paper waste and increasing resource efficiency has become a priority in the offset printing industry. We consider the production process of a company that prints customer-specific designs on napkin pouches. The planning problem consists of assigning the designs to slots of some printing plates such that the demand for each design is met, all technological constraints are satisfied, and the total production costs, which include setup and waste costs, are minimized. The main contribution of the paper is a matheuristic that takes advantage of the fact that the demands for individual designs are a multiple of a minimum order quantity. In a computational analysis based on real-world problem instances, we compared the proposed matheuristic to an exact approach. The matheuristic consistently outperformed the exact approach in terms of solution quality and running time.

Clustering addresses the problem of assigning similar objects to groups. Since the size of the clusters is often constrained in practical clustering applications, various capacitated clustering problems have received increasing attention. We consider here the capacitated p-median problem (CPMP) in which p objects are selected as cluster centers (medians) such that the total distance from these medians to their assigned objects is minimized. Each object is associated with a weight, and the total weight in each cluster must not exceed a given capacity. Numerous exact and heuristic solution approaches have been proposed for the CPMP. The state-of-the-art approach performs well for instances with up to 5,000 objects but becomes computationally expensive for instances with a much larger number of objects. We propose a matheuristic with new problem decomposition strategies that can deal with instances comprising up to 500,000 objects. In a computational experiment, the proposed matheuristic consistently outperformed the state-of-the-art approach on medium-and large-scale instances while having similar performance for small-scale instances. As an extension, we show that our matheuristic can be applied to related capacitated clustering problems, such as the capacitated centered clustering problem (CCCP). For several test instances of the CCCP, our matheuristic found new best-known solutions.

The k-means algorithm is one of the most widely-used algorithms in clustering. It is known to be effective when the clusters are homogeneous and well separated in the feature space. When this is not the case, incorporating pairwise must-link and cannot-link constraints can improve the quality of the resulting clusters. Various extensions of the k-means algorithm have been proposed that incorporate the must-link and cannot-link constraints using heuristics. We introduce a different approach that uses a new mixed-integer programming formulation. In our approach, the pairwise constraints are incorporated as soft-constraints that can be violated subject to a penalty. In a computational study based on 25 data sets, we compare the proposed algorithm to a state-of-the-art algorithm that was previously shown to dominate the other algorithms in this area. The results demonstrate that the proposed algorithm provides better clusterings and requires considerably less running time than the state-of-the-art algorithm. Moreover, we found that the ability to vary the penalty is beneficial in situations where the pairwise constraints are noisy due to corrupt ground truth.

We deal with the following application of the cardinality-constrained covering traveling salesperson problem. A company offers the valuation of real-estate properties, which includes an on-site visit by a contractor. Each contractor visits several properties during a tour, which must comprise not less than a minimum and not more than a maximum number of visits and must not exceed a prescribed length. Given a set of properties, the planning problem is to determine the respective tours such that the total relevant cost of all tours is minimized; for each tour, this cost consists of some fixed costs plus some variable costs proportional to the total distance of the tour. We propose a novel order-first split-second approach which at first devises a giant tour, then splits this tour into feasible tours, and eventually tries to improve these tours individually. Our computational results for a set of test instances from the literature indicate that the proposed approach runs much faster than the reference approaches and devises good feasible solutions; for the largest instances, the proposed approach even outperforms the reference approaches.

In this chapter, we propose a decision support system that allows analysts to assess the reliability of street networks. The system consists of three main components: a graph construction tool that transforms OpenStreetMap data into a directed graph, a traffic estimator that defines the traffic volume between origin-destination pairs, and an optimization model that determines an optimal flow of traffic from origins to destinations. We apply this system to the nation-wide street network of Switzerland. We also discuss how this system may be useful for the analysis of train networks, and we point to opportunities for future research.

Clustering is probably the most extensively studied problem in unsupervised learning. Traditional clustering algorithms assign objects to clusters exclusively based on features of the objects. Constrained clustering is a generalization of traditional clustering where additional information about a dataset is given in the form of constraints. It has been shown that the clustering accuracy can be improved substantially by accounting for these constraints. We consider the constrained clustering problem where additional information is given in the form of must-link and cannot-link constraints for some pairs of objects. Various algorithms have been developed for this specific clustering problem. We propose a binary linear programming-based k-means approach that can consider must-link and cannot-link constraints. In a computational experiment, we compare the proposed algorithm to the DILSCC algorithm, which represents the state-of-the-art. Our results on 75 problem instances indicate that the proposed algorithm delivers better clusterings than the DILSCC algorithm in much shorter running time.

Direct marketing has become a fundamental advertising method in many industries. In direct marketing, companies target specific customers with personalized product offers. By optimally assigning customers to direct marketing activities, the effectiveness of direct marketing campaigns can be greatly increased. In this paper, we study a real-world customer assignment problem of a leading telecommunications provider in Switzerland. The planning problem contains many business and customer-specific constraints that have not yet been covered in the literature. We propose a binary linear programming formulation that solves instances involving up to one million customers and over 100 direct marketing activities to optimality in short running time. The novel formulation delivers substantially better solutions in terms of expected profit than the current practice at the company.

The k-means algorithm is one of the most popular clustering algorithms in the machine learning community. Its simplicity and scalability make it the primary choice for many clustering applications. We introduce here a variant of the k-means algorithm that can account for complex side constraints. The key idea is to use binary linear programming for assigning objects to clusters. Unlike existing extensions of the k-means algorithm that are designed for accommodating specific types of constraints, our approach can be applied to a wide range of constrained clustering problems with minor modifications. We demonstrate the effectiveness and efficiency of the proposed approach by comparing it to a state-of-the-art algorithm on a test set that comprises 18 instances of the capacitated centered clustering problem. The proposed approach performed particularly well on large-sized instances with more than 100 clusters. It even found new best-known solutions for the four largest instances in the test set.

We present here a computational study comparing the performance of leading machine learning techniques to that of recently developed graph-based combinatorial optimization algorithms (SNC and KSNC). The surprising result of this study is that SNC and KSNC consistently show the best or close to best performance in terms of their F-1-scores, accuracy, and recall. Furthermore, the performance of SNC and KSNC is considerably more robust than that of the other algorithms; the others may perform well on average but tend to vary greatly across data sets. This demonstrates that combinatorial optimization techniques can be competitive as compared to state-of-the-art machine learning techniques. The code developed for SNC and KSNC is publicly available. (C) 2018 Elsevier B.V. All rights reserved.

Significant long-term cost savings can be achieved when labor-intensive daily operations are executed at minimal cost. We consider here a real-world planning problem that was reported to us by a real estate valuation company. The planning problem consists of scheduling on-site visits such that the total operating costs are minimized. We show that this problem represents a new variant of the multiple traveling salesman problem to which existing approaches cannot be applied directly. We formulate the problem as a mixed-binary linear program and develop a matheuristic for large-scale instances. The matheuristic employs a new strategy to construct subproblems effectively and techniques to exclude variables that are unlikely to be non-zero in an optimal solution. Our computational analysis demonstrates that the mixed-binary linear program is able to devise optimal or near-optimal solutions for instances with up to 200 visits in short running times. The matheuristic performs equally well on smalland medium-sized instances and proves to be highly scalable.