Set Partitioning Research Articles

Set partitions avoid a four-letter pattern

We present an explicit formula for the generating function for the number of set partitions of {1,2,…,n} that avoid a four-letter pattern 1abc.

Determining [formula omitted]- and ([formula omitted], [formula omitted])-robustness of multiagent networks based on heuristic algorithm

The graph properties of r- and (r,s)-robustness are of importance for multiagent networks since these properties ensure consensus among agents even in the presence of a limited number of arbitrarily misbehaving agents, given sufficiently large values for the integers r and s. However, determining the exact solutions of r- and (r,s)-robustness of an arbitrary directed graph has been proven to be computationally complex NP-hard problem. In this paper, we introduce a novel method, named the Determining Robustness based Genetic Algorithm (DRSGA), for approximately calculating the r- and (r,s)-robustness using heuristic algorithm. According to graph theory in mathematical analysis, we first formalize the method for calculating the r- and (r,s)-robustness in directed graphs by utilizing a three-vertex set partition. Then, we transform these methods into a minimization problem of an n-dimensional discrete function and employ DRSGA to obtain an approximate solution. Finally, we validate the efficacy of our algorithm through a series of experiments compared to existing mixed-integer programming algorithms.

An improved genetic algorithm based on reinforcement learning for aircraft assembly scheduling problem

The assembly work of aircraft occupies a significant portion of the time in aircraft manufacturing. The aircraft assembly operations are highly challenging in the generation of production plans due to multiple constraints in the scheduling process. In addressing the aircraft assembly scheduling problem with the objective of minimizing the duration of assembly operations, a mathematical model has been established. This model incorporates characteristics such as job sequencing constraints, resource constraints, spatial constraints, and others, the problem-solving process is divided into two stages: determining the priority sequence of tasks and establishing the start times for each task. Proposing an improved genetic algorithm based on Q-learning, referred to as QIGA (Q-learning-based Improved Genetic Algorithm), utilizing Q-learning to dynamically adjust the crossover and mutation probability parameters during each iteration to enhance the algorithm’s convergence speed and search capabilities. Constructing a Markov Decision Process (MDP) model to dynamically adjust the crossover and mutation probability in the problem, proposing a state set partition rule based on the population’s fitness values, setting the action set to represent crossover and mutation probabilities in different intervals. Additionally, designing reward rules based on the population’s performance. Finally, the effectiveness of the proposed algorithm is verified with a real case containing 44 assembly operations, and the results show that the algorithm can solve the aircraft assembly scheduling problem well. Validating the feasibility and optimization effectiveness of the algorithm through various aircraft assembly scheduling cases of different scales. The experimental results indicate that, compared to traditional meta-heuristic algorithms, the proposed algorithm demonstrates superior optimization performance in aircraft assembly scheduling problems. It effectively reduces the total duration of aircraft assembly operations.

Growth period determination and color coordinates visual analysis of tomato using hyperspectral imaging technology

Growth period determination and color coordinates prediction are essential for comparing postharvest fruit quality. This paper proposes a tomato growth period judgment and color coordinates prediction model based on hyperspectral imaging technology. It utilizes the most effective color coordinates prediction model to obtain a color visual image. Firstly, hyperspectral images were taken of tomatoes at different growth periods (green-ripe, color-changing, half-ripe, and full-ripe), and color coordinates (L*, a*, b*, c, h) were obtained using a colorimeter. The sample set was divided by the sample set partitioning based on joint X-Y distances (SPXY). The support vector machine (SVM), K-nearest neighbors (KNN), and linear discriminant analysis (LDA) were used to discriminate growth period. Results show that the LDA model has the best prediction effect with a prediction set accuracy of 93.1%. In addition, effective wavelengths were selected using competitive adaptive reweighted sampling (CARS) and successive projections algorithm (SPA), and chromaticity prediction models were established using partial least squares regression (PLSR), multiple linear regression (MLR), principal component regression (PCR) and support vector machine regression (SVR) Finally, the color of each pixel of the tomato is calculated using the optimal model, generating a visual distribution image of the color coordinate. The results showed that hyperspectral imaging can non-destructively detect tomatoes’ growth stage and color coordinates, providing great significance for designing a tomato quality grading system.

Ensemble multi-view feature set partitioning method for effective multi-view learning

Ensemble multi-view feature set partitioning method for effective multi-view learning

Prediction of Student Dropout in Malaysian’s Private Higher Education Institute using Data Mining Application

Student dropout issue is a major concern among the academics and management of the university. The higher rate of student dropout impacted the university reputation such as reducing student enrolment, affecting the revenue of the university, financial losses for the country, and increase the existence of a social problem among the students. In this study, 2 popular classifiers were utilized to predict the student dropout namely decision tree and logistic regression model respectively. Several sets of experimental setting were employed which include three set of data partitioning - along with different types of decision tree and regression model. As for the logistic regression model, different data imputation and transformation method was tested to ensure that the model built is valid. A total of 7706 student data extracted from one of the private universities in Malaysia database (between year 2018-2019) to assess the capability of the classifier. The classifier performance is evaluated using machine learning performance measure of accuracy and misclassification rate. The result indicates that, decision tree - chi-square (2 branches) achieved slightly better classification performance of 89.49% on 80/20 data partitioning. The chosen model also identified the most important variable for accurate prediction of student dropout. Application of this model has the potential to accurately predict at risk student and to reduce student dropout rates.

Non-Destructive Detection of Cerasus Humilis Fruit Quality by Hyperspectral Imaging Combined with Chemometric Method

Cerasus Humilis fruit is susceptible to rapid color changes post-harvest, which degrades its quality. This research utilized hyperspectral imaging technology to detect and visually analyze the soluble solid content (SSC) and firmness of the fruit, aiming to improve quality and achieve optimal pricing. Four maturity stages (color turning stage, coloring stage, maturity stage, and fully ripe stage) of Cerasus Humilis fruit were examined using hyperspectral images (895–1700 nm) alongside data collection on SSC and firmness. These samples were divided into a calibration set and a validation set with a ratio of 3:1 by sample set partitioning based on the joint X-Y distances (SPXY) method. The original spectral data was processed by a spectral preprocessing method. Multiple linear regression (MLR) and nonlinear least squares support vector machine (LS-SVM) detection models were established using feature wavelengths selected by the successive projections algorithm (SPA), competitive adaptive reweighted sampling (CARS), uninformative variable elimination (UVE), and two combined downscaling algorithms (UVE-SPA and UVE-CARS), respectively. For SSC and firmness detection, the best models were the SNV-SPA-LS-SVM model with 18 feature wavelengths and the original spectra-UVE-CARS-LS-SVM model with eight feature wavelengths, respectively. For SSC, the correlation coefficient of prediction (Rp) was 0.8526, the root mean square error of prediction (RMSEP) was 0.9703, and the residual prediction deviation (RPD) was 1.9017. For firmness, Rp was 0.7879, RMSEP was 1.1205, and RPD was 2.0221. Furthermore, the optimal model was employed to retrieve the distribution of SSC and firmness within Cerasus Humilis fruit. This retrieved information facilitated visual inspection, enabling a more intuitive and comprehensive assessment of SSC and firmness at each pixel level. These findings demonstrated the effectiveness of hyperspectral imaging technology for determining SSC and firmness in Cerasus Humilis fruit. This paves the way for online monitoring of fruit quality, ultimately facilitating timely harvesting.

An enhanced exact algorithm for the multi-trip vehicle routing problem with time windows and capacitated unloading station

This paper introduces a vehicle routing problem that simultaneously considers multiple trips, time windows, and a capacitated unloading station. This problem is a generalization of the multi-trip vehicle routing problem with time windows, which determines a set of least-cost vehicle routes to fulfill all customer demands while respecting the constraints of vehicle capacity and time windows. Due to restricted resources (e.g., equipment and labor force) at the depot, vehicles may need to wait in a queue for being unloaded when they arrive. This unloading capacity constraint significantly complicates the problem, as it causes a trip to involve three stages—traveling, waiting, and unloading. We formulate this problem as an arc flow model and a trip-based set partitioning model, where the latter is solved by a branch-price-and-cut (BPC) algorithm. To improve the computational aspect of the BPC framework, a two-phase column generation (CG) algorithm is designed. First, a bidirectional labeling algorithm is tailored to solve the pricing problem, where two accelerating strategies are employed to speed up the resolution process. Meanwhile, k-path inequalities and limited-memory subset row inequalities are utilized to tighten the linear relaxation of the master problem. Computational results based on the instances adapted from the well-known Solomon’s benchmark show that the developed BPC algorithm can solve most instances within 50 customers to optimality in a short time frame and some instances of 100 customers to optimality within a 3-hour time limit. Moreover, our BPC algorithm performs better than exact algorithms in the literature for similar problem variants in both solution quality and computing time.

BGS: Accelerate GNN training on multiple GPUs

Emerging Graph Neural Networks (GNNs) have made significant progress in processing graph-structured data, yet existing GNN frameworks face scalability issues when training large-scale graph data using multiple GPUs. Frequent feature data transfers between CPUs and GPUs are a major bottleneck, and current caching schemes have not fully considered the characteristics of multi-GPU environments, leading to inefficient feature extraction. To address these challenges, we propose BGS, an auxiliary framework designed to accelerate GNN training from a data perspective in multi-GPU environments. Firstly, we introduce a novel training set partition algorithm, assigning independent training subsets to each GPU to enhance the spatial locality of node access, thus optimizing the efficiency of the feature caching strategy. Secondly, considering that GPUs can communicate at high speeds via NVLink connections, we designed a feature caching placement strategy suitable for multi-GPU environments. This strategy aims to improve the overall hit rate by setting reasonable redundant caches on each GPU. Evaluations on two representative GNN models, GCN and GraphSAGE, show that BGS significantly improves the hit rate of feature caching strategies in multi-GPU environments and substantially reduces the time overhead of data loading, achieving a performance improvement of 1.5 to 6.2 times compared to the baseline.

Generation of PN-Model Synthesis Programs with Restrictions

The paper proposes a modified algorithm for generating programs for the synthesis of network models expressed in terms of Petri nets [8]. The algorithm is based on the use of boundedly growing strings and the ability to generate them in lexicographic order. To reduce the number of possible synthesis programs, the introduction of limiting conditions imposed on the set partition tree is considered. Taking into account the imposed restrictions, a new tree of smaller dimension is built, which also preserves the lexicographic order of the limitedly growing strings it contains. A comparative assessment of the effectiveness of a new algorithm for generating boundedly growing strings with restrictions and a standard algorithm without restrictions, which is described by D. Knuth, is given. The proposed algorithm can be used to solve problems of synthesizing models of complex computational structures, building parallel systems, parallelizing algorithms and solving other problems related to the creation of efficient distributed systems.

The asymptotics of tensor powers of Foulkes modules

In this paper we study the modular structure of the Foulkes module H ( 2 m ) of the symmetric group S 2 m acting on set partitions of a set of size 2m into m sets each of size 2, defined over a field of positive characteristic p. We aim to understand the decomposition of the tensor powers of H ( 2 m ) . In particular, we study the asymptotics of the non-projective part of tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds.

The static ridesharing routing problem with flexible locations: A Norwegian case study

The municipalities in the Bergen region in Norway have recently announced a pilot project for ridesharing in the region as a means to reduce traffic congestion. As part of this project, we study the Static Ridesharing Routing Problem with Flexible Locations (SRRPFL), which aims at determining efficient routes and schedules for a set of drivers to pick up and deliver passengers at different, flexible pickup and delivery locations. We present a bi-objective mixed integer programming (MIP) model for the SRRPFL where we (lexicographically) first maximize the number of passengers serviced and then minimize the total travel times. To solve real-life instances of the SRRPFL, we propose a new Adaptive Large Neighborhood Search (ALNS) heuristic. To further improve its performance, we extend the ALNS heuristic with a local search, as well as with a set partitioning problem (denoted the Route Combination Problem) that optimally recombines the routes previously encountered in the search. The ALNS heuristic is tested on a number of test instances based on real trip data and the results demonstrate its effectiveness. The results also provide a number of insights regarding the potential benefits of ridesharing in our case study.

Parsimonious consensus hierarchies, partitions and fuzzy partitioning of a set of hierarchies

Methodology is described for fitting a fuzzy partition and a parsimonious consensus hierarchy (ultrametric matrix) to a set of hierarchies of the same set of objects. A model defining a fuzzy partition of a set of hierarchical classifications, with every class of the partition synthesized by a parsimonious consensus hierarchy is described. Each consensus includes an optimal consensus hard partition of objects and all the hierarchical agglomerative aggregations among the clusters of the consensus partition. The performances of the methodology are illustrated by an extended simulation study and applications to real data. A discussion is provided on the new methodology and some interesting future developments are described.

On Fall-Colorable Graphs

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set V(G) in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least (n(k−1)+r(k−r))/2, where r≡n(modk) and 0≤r≤k−1, and the bound is tight. Then, we obtain that if G is k-colorable (k≥2) and the minimum degree of G is at least k−2k−1n, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k≥3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k∈{5,6}.

VLSI design for adjustable compression rate in lossless/lossy compression of EEG signal

In the realm of remote health monitoring, the compression of electroencephalography (EEG) signals at the edge holds significant importance. This paper presents a VLSI structure capable of adjustable compression rates to achieve both lossless and lossy compression of EEG data, catering to environments requiring flexible adjustment of compression types and rates. The architecture integrates two-dimensional discrete wavelet transform (2D-DWT) and the Set Partitioning in Hierarchical Trees (SPIHT). Notably, the Haar wavelet lifting structure is employed in 2D-DWT, and the SPIHT scanning sequence is optimized for enhanced resource efficiency. The proposed algorithm is validated using the CHB-MIT Scalp EEG Database, demonstrating compression ratios of ≥1.95 for normal EEG and ≥1.69 for epileptic EEG. Through synthesis and layout design in the 130 nm CMOS technology, the resulting circuit achieves a maximum clock frequency of 57 MHz and occupies an area of 81 kμm2, with an average power consumption of less than 0.71nj per data for 16-bit data. These findings affirm the efficacy of the proposed VLSI structure for achieving efficient lossless/lossy compression of edge EEG data.

МАТЕМАТИЧНІ МОДЕЛІ ТА МЕТОДИ РОЗМІЩЕННЯ ОБ’ЄКТІВ І ЗОНУВАННЯ ТЕРИТОРІЙ В СИСТЕМАХ ЕКСТРЕНОЇ ЛОГІСТИКИ

The mathematical models for distribution processes related to organizing precautionary measures in the event of threats or occurrences of man-made emergencies are presented. The tasks include optimal zoning of territories with the fixing of zones by objects of social purpose for service provision. Provision is made for: the possibility of overlapping zones in case the nearest center cannot provide the service; optimal placement of a certain number of new cen-ters of emergency logistics systems with simultaneous redistribution of the load on all their structural elements; the selection of locations of structural subdivisions based on existing fa-cilities. The optimality criteria involve minimizing either the time to provide the service even to the most remote object in the given territory, or the total distance to the nearest centers from consumers that are densely distributed in the given territory, and/or the organizational costs associated with the arrangement of new centers. Mathematical models are proposed in the form of continuous problems of optimal multiplex partitioning of sets with a linear or minimax functional of quality. The latter provides such placement of centers that provides op-timal multiple coverage of the territory (with a minimum radius of multiple coverage). Meth-ods for solving the formulated problems were developed using LP-relaxation of linear prob-lems with Boolean variables, duality theory to reduce the initial problems of infinite-dimensional programming to problems of conditional optimization of a non-smooth function of several variables, and modern methods of non-differentiated optimization.

Locating Chromatic Number of Middle Graph of Path, Cycle, Star, Wheel, Gear and Helm Graphs

Let \(c\) be a proper \(k\)-coloring of a connected graph \(G\) and \(\pi=\{S_{1},S_{2},\ldots,S_{k}\}\) be an ordered partition of the vertex set \(V(G)\) into the resulting color classes, where \(S_{i}\) is the set of all vertices that receive the color \(i\). For a vertex \(v\) of \(G\), the color code \(c_{\pi}(v)\) of \(v\) with respect to \(\pi\) is the ordered \(k\)-tuple \(c_{\pi}(v)=(d(v,S_{1}),d(v,S_{2}),\ldots,d(v,S_{k}))\), where \(d(v,S_{i})=min\{d(v,u):\textit{ } u\in S_{i}\}\) for \(1\leqslant i \leqslant k\). If all distinct vertices of \(G\) have different color codes, then \(c\) is called a locating coloring of \(G\). The locating chromatic number is the minimum number of colors needed in a locating coloring. In this paper, we determine the locating-chromatic number for the middle graphs of Path, Cycle, Wheel, Star, Gear and Helm graphs.

An Effective Two-Stage Algorithm for the Bid Generation Problem in the Transportation Service Market

This study designs a two-stage algorithm to address the bid generation problem of carriers when adding new vehicle routes in the presence of the existing vehicle routes to provide transportation service. To obtain the best auction combination and bid price of the carrier, a hybrid integer nonlinear programming model is introduced. According to the characteristics of the problem, a set of two-stage hybrid algorithms is proposed, innovatively integrating block coding within a genetic algorithm framework with a depth-first search approach. This integration effectively manages routing constraints, enhancing the algorithm’s efficiency. The block coding and each route serve as decision variables in the set partition formula, enabling a comprehensive exploration of potential solutions. After a simulation-based analysis, the algorithm has been comprehensively validated analytically and empirically. The improvement of this research lies in the effectiveness of the proposed algorithm, i.e., the ability to handle a broader range of problem scales with less time in addressing complex operator bid generation in combinatorial auctions.

A Note on Internal Partitions: The 5-Regular Case and Beyond

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the 5-regular case and show that among the subgraphs of minimum degree at least 3 of 5-regular graphs, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all 5-regular Abelian Cayley graphs which do not have an internal partition.

New Results on the Robust Coloring Problem

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called equitable coloring. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.