Set Partitioning Research Articles

Solving a real-world multi-depot multi-period petrol replenishment problem with complex loading constraints

In this paper, we address a new variant of the petrol replenishment problem (PRP), which is a rich real-word multi-depot multi-period problem (MDMPPRP). We show that it is possible to solve this complex variant with an exact branch-and-price approach and some derived heuristics. On one side, this problem could be modeled as a set partitioning type problem with low to moderate density (the number of ones per column, i.e., clients to visit, is not large). Such problems have some nice polyhedral properties to consider for favoring integrality. In the other side, some complex handling rules apply due to the problem’s context. A natural way is to address them in the column generation subproblem as an elementary shortest path problem with resource constraints, which constitutes the major bottleneck. To succeed in this challenge, we need to design some sophisticated techniques i) for branching to profit from the polyhedral properties and ii) for solving the column generation subproblem. Direct use of on-the-shelf algorithms does not work, unfortunately. Numerical results on a real network (four depots, five types of petroleum products, four main groups of clients, heterogeneous fleet of highly compartmented tank trucks) prove the effectiveness and high potential of the proposed approach.

Seaweeds Arising from Brauer Configuration Algebras

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra Mat(n) introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting that finding such an index for general Lie algebras is a cumbersome problem. On the other hand, Brauer configuration algebras are multiserial and symmetric algebras whose representation theory can be described using combinatorial data. It is worth pointing out that the set of integer partitions and compositions of a fixed positive integer give rise to Brauer configuration algebras. However, giving a closed formula for the dimension of these kinds of algebras or their centers for all positive integer is also a tricky problem. This paper gives formulas for the dimension of Brauer configuration algebras (and their centers) induced by some restricted compositions. It is also proven that some of these algebras allow defining seaweeds of Dixmier index one.

Partition of regular balanced c-tournaments into strongly connected c-tournaments

A multipartite tournament will be called r-balanced if every partite set has r vertices. The study of the existence of strongly connected subtournaments in multipartite tournaments started in 1999 by Volkmann. In this paper we study a much more general problem: the existence of a partition of a c-partite tournament into strongly connected tournaments of order c, which will be called strong partition. Given a positive integer r, the strong partition number, ST(r), is the minimum integer c′ such that every regular r-balanced c-partite tournament, with c≥c′, has a strong partition. We prove that for every r≥2, ST(r) exists and that 5≤ST(2)≤7. Moreover, as a consequence of our technique, for every r, there exists a c such that for every c′>c, there exists a strong partition into maximal tournaments of minimum degree ⌊c′−24⌋+1.

Enumeration of symmetric arc diagrams

We give recurrence relations for the enumeration of symmetric elements within four classes of arc diagrams corresponding to certain involutions and set partitions whose blocks contain no consecutive integers. These arc diagrams are motivated by the study of RNA secondary structures. For example, classic RNA secondary structures correspond to 3412-avoiding involutions with no adjacent transpositions, and structures with base triples may be represented as partitions with crossings. Our results rely on combinatorial arguments. In particular, we use Motzkin paths to describe noncrossing arc diagrams that have no arc connecting two adjacent nodes, and we give an explicit bijection to ternary words whose length coincides with the sum of their digits. We also discuss the asymptotic behavior of some of the sequences considered here in order to quantify the extremely low probability of finding symmetric structures with a large number of nodes.

Generalized Cut Method for Computing Szeged–Like Polynomials with Applications to Polyphenyls and Carbon Nanocones

Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On the other hand, the polynomials related to these topological indices were also introduced, for example the Szeged polynomial, the edgeSzeged polynomial, the PI polynomial, the Mostar polynomial, etc. In this paper, we introduce a concept of the general Szeged-like polynomial for a connected strength-weighted graph. It turns out that this concept includes all the above mentioned polynomials and also infinitely many other graph polynomials. As the main result of the paper, we prove a cut method which enables us to efficiently calculate a Szeged-like polynomial by using the corresponding polynomials of strength-weighted quotient graphs obtained by a partition of the edge set that is coarser than Θ∗ -partition. To the best of our knowledge, this represents the first implementation of the famous cut method to graph polynomials. Finally, we show how the deduced cut method can be applied to calculate some Szeged-like polynomials and corresponding topological indices of para-polyphenyl chains and carbon nanocones.

Routing Optimization with Vehicle–Customer Coordination

In several transportation systems, vehicles can choose where to meet customers rather than stopping in fixed locations. This added flexibility, however, requires coordination between vehicles and customers that adds complexity to routing operations. This paper develops scalable algorithms to optimize these operations. First, we solve the one-stop subproblem in the [Formula: see text] space and the [Formula: see text] space by leveraging the geometric structure of operations. Second, to solve a multistop problem, we embed the single-stop optimization into a tailored coordinate descent scheme, which we prove converges to a global optimum. Third, we develop a new algorithm for dial-a-ride problems based on a subpath-based time–space network optimization combining set partitioning and time–space principles. Finally, we propose an online routing algorithm to support real-world ride-sharing operations with vehicle–customer coordination. Computational results show that our algorithm outperforms state-of-the-art benchmarks, yielding far superior solutions in shorter computational times and can support real-time operations in very large-scale systems. From a practical standpoint, most of the benefits of vehicle–customer coordination stem from comprehensively reoptimizing “upstream” operations as opposed to merely adjusting “downstream” stopping locations. Ultimately, vehicle–customer coordination provides win–win–win outcomes: higher profits, better customer service, and smaller environmental footprint. This paper was accepted by Chung Piaw Teo, optimization. Funding: This research was supported by the National Natural Science Foundation of China [Grants 72288101, 52221005 and 52220105001]. Supplemental Material: The e-companion and data are available at https://doi.org/10.1287/mnsc.2023.4739 .

Methodological Issues in Evaluating Machine Learning Models for EEG Seizure Prediction: Good Cross-Validation Accuracy Does Not Guarantee Generalization to New Patients

There is an increasing interest in applying artificial intelligence techniques to forecast epileptic seizures. In particular, machine learning algorithms could extract nonlinear statistical regularities from electroencephalographic (EEG) time series that can anticipate abnormal brain activity. The recent literature reports promising results in seizure detection and prediction tasks using machine and deep learning methods. However, performance evaluation is often based on questionable randomized cross-validation schemes, which can introduce correlated signals (e.g., EEG data recorded from the same patient during nearby periods of the day) into the partitioning of training and test sets. The present study demonstrates that the use of more stringent evaluation strategies, such as those based on leave-one-patient-out partitioning, leads to a drop in accuracy from about 80% to 50% for a standard eXtreme Gradient Boosting (XGBoost) classifier on two different data sets. Our findings suggest that the definition of rigorous evaluation protocols is crucial to ensure the generalizability of predictive models before proceeding to clinical trials.

On the number of vertices of projective polytopes

AbstractLet X be a set of n points in in general position. What is the maximum number of vertices that can have among all the possible permissible projective transformations T? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related problem (via Gale transforms) concerning the maximal number of minimal Radon partitions of a set of points. The latter led us to a result supporting a positive answer to a question of Pach and Szegedy asking whether balanced 2‐colorings of points in the plane maximize the number of induced multicolored Radon partitions. We also discuss a related problem concerning the size of topes in arrangements of hyperplanes as well as a tolerance‐type problem of finite sets.

Complexity Analysis of Benes Network and Its Derived Classes via Information Functional Based Entropies

The use of information–theoretical methodologies to assess graph-based systems has received a significant amount of attention. Evaluating a graph’s structural information content is a classic issue in fields such as cybernetics, pattern recognition, mathematical chemistry, and computational physics. Therefore, conventional methods for determining a graph’s structural information content rely heavily on determining a specific partitioning of the vertex set to obtain a probability distribution. A network’s entropy based on such a probability distribution is obtained from vertex partitioning. These entropies produce the numeric information about complexity and information processing which, as a consequence, increases the understanding of the network. In this paper, we study the Benes network and its novel-derived classes via different entropy measures, which are based on information functionals. We construct different partitions of vertices of the Benes network and its novel-derived classes to compute information functional dependent entropies. Further, we present the numerical applications of our findings in understanding network complexity. We also classify information functionals which describe the networks more appropriately and may be applied to other networks.

Optimal unlabeled set partitioning with application to risk-based quarantine policies

We consider the problem of partitioning a set of items into unlabeled subsets so as to optimize an additive objective, i.e., the objective function value of a partition is equal to the sum of the contribution of each subset. Under an arbitrary objective function, this family of problems is known to be an -complete combinatorial optimization problem. We study this problem under a broad family of objective functions characterized by elementary symmetric polynomials, which are “building blocks” to symmetric functions. By analyzing a continuous relaxation of the problem, we identify conditions that enable the use of a reformulation technique in which the set partitioning problem is cast as a more tractable network flow problem solvable in polynomial-time. We show that a number of results from the literature arise as special cases of our proposed framework, highlighting its generality. We demonstrate the usefulness of the developed methodology through a novel and timely application of quarantining heterogeneous populations in an optimal manner. Our case study on real COVID-19 data reveals significant benefits over conventional measures in terms of both spread mitigation and economic impact, underscoring the importance of data-driven policies.

Berkovich–Uncu Type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect $$t^{th}$$ power. In addition, the partitions that we study here have smallest part greater than or equal to s for some given natural number s. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case $$t=1$$ . To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of $$\mathbb {N}^t$$ , and Jensen’s inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.

Problem-driven scenario clustering in stochastic optimization

In stochastic optimisation, the large number of scenarios required to faithfully represent the underlying uncertainty is often a barrier to finding efficient numerical solutions. This motivates the scenario reduction problem: by finding a smaller subset of scenarios, reduce the numerical complexity while keeping the error at an acceptable level. In this paper we propose a novel and computationally efficient methodology to tackle the scenario reduction problem for two-stage problems when the error to be minimised is the implementation error, i.e. the error incurred by implementing the solution of the reduced problem in the original problem. Specifically, we develop a problem-driven scenario clustering method that produces a partition of the scenario set. Each cluster contains a representative scenario that best reflects the optimal value of the objective function in each cluster of the partition to be identified. We demonstrate the efficiency of our method by applying it to two challenging two-stage stochastic combinatorial optimization problems: the two-stage stochastic network design problem and the two-stage facility location problem. When compared to alternative clustering methods and Monte Carlo sampling, our method is shown to clearly outperform all other methods.

Keep your distance: Land division with separation

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

A combinatorial basis for the fermionic diagonal coinvariant ring

Let \(\Theta_n = (\theta_1, \dots, \theta_n)\) and \(\Xi_n = (\xi_1, \dots, \xi_n)\) be two lists of \(n\) variables and consider the diagonal action of \(\mathfrak{S}_n\) on the exterior algebra \(\wedge \{ \Theta_n, \Xi_n \}\) generated by these variables. Jongwon Kim and Rhoades defined and studied the fermionic diagonal coinvariant ring \(FDR_n\) obtained from \(\wedge \{ \Theta_n, \Xi_n \}\) by modding out by the \(\mathfrak{S}_n\)-invariants with vanishing constant term. The author and Rhoades gave a basis for the maximal degree components of this ring where the action of \(\mathfrak{S}_n\) could be interpreted combinatorially via noncrossing set partitions. This paper will do similarly for the entire ring, although the combinatorial interpretation will be limited to the action of \(\mathfrak{S}_{n-1} \subset \mathfrak{S}_n\). The basis will be indexed by a certain class of noncrossing partitions.Mathematics Subject Classifications: 05E10, 05E18, 20C30Keywords: Skein relation, coinvariant algebra, noncrossing set partition, cyclic sieving

The Haglund–Remmel–Wilson identity for set partitions

In 2015, Remmel and Wilson proved the following conjecture of Jim Haglund for permutations∑π∈Snqinv(π)∏j∈Des(π)(1+zq1+inv□,j(π))=∑π∈Snqmaj(π)∏j=1des(π)(1+zqj). Wilson extended this identity to words in 2016. In this note, we prove that this identity holds for set partitions.

Machine-learning-based head impact subtyping based on the spectral densities of the measurable head kinematics

BackgroundTraumatic brain injury can be caused by head impacts, but many brain injury risk estimation models are not equally accurate across the variety of impacts that patients may undergo, and the characteristics of different types of impacts are not well studied. We investigated the spectral characteristics of different head impact types with kinematics classification. MethodsData were analyzed from 3262 head impacts from lab reconstruction, American football, mixed martial arts, and publicly available car crash data. A random forest classifier with spectral densities of linear acceleration and angular velocity was built to classify head impact types (e.g., football, car crash, mixed martial arts). To test the classifier robustness, another 271 lab-reconstructed impacts were obtained from 5 other instrumented mouthguards. Finally, with the classifier, type-specific, nearest-neighbor regression models were built for brain strain. ResultsThe classifier reached a median accuracy of 96% over 1000 random partitions of training and test sets. The most important features in the classification included both low- and high-frequency features, both linear acceleration features and angular velocity features. Different head impact types had different distributions of spectral densities in low- and high-frequency ranges (e.g., the spectral densities of mixed martial arts impacts were higher in the high-frequency range than in the low-frequency range). The type-specific regression showed a generally higher R2 value than baseline models without classification. ConclusionThe machine-learning-based classifier enables a better understanding of the impact kinematics spectral density in different sports, and it can be applied to evaluate the quality of impact-simulation systems and on-field data augmentation.

Robust ML model ensembles via risk-driven anti-clustering of training data

In this paper, we improve the robustness of Machine Learning (ML) classifiers against training-time attacks by linking the risk of training data being tampered with to the redundancy in the ML model's design needed to prevent it. Our defense mechanism is directly applicable to classifiers' training data, without any knowledge of the specific ML model to be hardened. First, we compute the training data proximity to class separation surfaces, identified via a reference linear model. Each data point is associated with a risk index, which is used to partition the training set by an unsupervised technique. Then, we train a learner for each partition and combine the learners' output in an ensemble. Our method treats the protected ML classifier as a black box and is inherently robust to transfer attacks. Experiments show that, for data poisoning rates between 6 and 25 percent of the training set, our method is more robust compared to benchmarks and to a monolithic version of the model trained on the whole training set. Our results make a convincing case for adopting training set partitioning and ensemble generation as a stage of ML models' development and deployment lifecycle.

Counting occurrences of subword patterns in non-crossing partitions

A permutation pattern in which all letters within an occurrence are required to be adjacent is known as a subword. In this paper, we consider the distribution of several infinite families of subword patterns on the set of non-crossing partitions of size n, denoted by NCn, and derive formulas for the generating functions of these distributions on NCn. As special cases of our results, we obtain formulas for the generating functions enumerating the members of NCn according to the number of occurrences of any subword of length three with distinct letters. Simple expressions for the total number of occurrences of a pattern over all members of NCn are also deduced. Some connections are made with the related problem of counting Dyck paths according to the number of occurrences of certain types of strings. Further, for the subwords 12⋯m and 213⋯m where m ≥ 3, we consider the joint distribution with the descents statistic and make use of the kernel method to establish the results in these cases.

On the multi-period combined maintenance and routing optimisation problem

This paper focuses on the combined maintenance and routing optimisation problem in a multi-period environment that consists of scheduling maintenance operations for a set of geographically distributed machines, subject to non-deterministic failures with a set of technicians that perform preventive maintenance and corrective operations. This study uses a two-step approach based on column generation. The first step consists of a maintenance model that determines the optimal time until the next preventive maintenance operation, and each machine’s maintenance frequency while minimising the total expected maintenance costs. The second step involves a routing model that assigns and schedules maintenance operations for each technician over the planning horizon while minimising the routing and maintenance costs. We formulate the second step problem as a periodic vehicle routing problem and to solve it, we propose a column generation approach. The master problem is a set partitioning formulation that indicates which machines must be visited in the planning horizon. This decomposition has two auxiliary problems: the first one is an elementary shortest-path problem to find a feasible path for each period and the second one finds a set of visiting periods for each machine. Our approach balances the maintenance cost, routing cost, and failure probabilities.

A coalition formation framework of smallholder farmers in an agricultural cooperative

Agricultural cooperatives remain a significant component of the food and agriculture industry to help the stakeholders to provide services and have opportunities for themselves. One of the aims of an agricultural cooperative is to answer to the needs within the communities of the farmers. Agricultural cooperatives enable individual farmers to increase productivity and maximise their social welfare. Together the farmer members of an agricultural cooperative can buy input supplies cheaper and sell more of their products in larger markets at higher prices, which is not possible for an individual smallholder farmer otherwise. Some studies have shown that farmers who were members of cooperatives have gained higher revenue for their products and spent less on input. However, organising the hundreds of farmers into smaller groups to perform collective farming and marketing is crucial to strengthening their position in the food and agriculture industry. Thereby, in our work, we consider an agricultural cooperative of smallholder farmers as a multi-agent based coalitional model, where coalitions are formed based on the similarity among the smallholder farmers. In this paper, we propose a model and implement a heuristic-based algorithm to find the disjoint partition of the agents set. We evaluate the model and the algorithm based on the following criteria: (i) individual gain, (ii) runtime analysis, (iii) solution quality, and (iv) scalability. We theoretically prove that our coalitional model of an agricultural cooperative has conciseness, expressiveness and efficiency properties. Experimental results confirm that our algorithm is time efficient and scalable. We show, both empirically and theoretically, that our algorithm generates a solution within a bound of the optimal solution. We also show that our coalition model generates positive revenue for the smallholder farmers and the payoff division rule is individual rational. In addition, we generate a new dataset in the context of an agricultural cooperative to show the effectiveness and efficiency of the proposed coalitional model of the cooperative.