Set Partitioning Research Articles

Identification of dominant propagation paths based on sub-synchronous oscillation using branch oscillation energy distribution coefficient

The large-scale integration of wind power into the power grid can cause a new type of sub-synchronous power oscillation, different from traditional thermal power generation. The oscillation energy will spread extensively in the grid, causing power oscillation and even grid-cascading events. To address this issue, this article proposes a method for quantitatively analyzing the propagation characteristics of oscillation energy based on branch oscillation energy. Firstly, analyzing the oscillation energy shared by different branches in the network based on transient energy function. Next, a method is proposed to identify the dominant propagation path of sub-synchronous oscillation by defining the oscillation energy of branches under the dominant oscillation mode and the oscillation energy distribution coefficient of each branch. The oscillation partition set formed by the dominant propagation path can be used to locate the high-risk oscillation area of the system. Finally, the effectiveness of the method proposed in this paper for studying the wide-area propagation characteristics of sub-synchronous oscillations was verified through time-domain simulation analysis.

BUS-BRA: A breast ultrasound dataset for assessing computer-aided diagnosis systems.

Computer-aided diagnosis (CAD) systems on breast ultrasound (BUS) aim to increase the efficiency and effectiveness of breast screening, helping specialists to detect and classify breast lesions. CAD system development requires a set of annotated images, including lesion segmentation, biopsy results to specify benign and malignant cases, and BI-RADS categories to indicate the likelihood of malignancy. Besides, standardized partitions of training, validation, and test sets promote reproducibility and fair comparisons between different approaches. Thus, we present a publicly available BUS dataset whose novelty is the substantial increment of cases with the above-mentioned annotations and the inclusion of standardized partitions to objectively assess and compare CADsystems. The BUS dataset comprises1875 anonymized images from1064 female patients acquired via four ultrasound scanners during systematic studies at the National Institute of Cancer (Rio de Janeiro, Brazil). The dataset includes biopsy-proven tumors divided into722 benign and342 malignant cases. Besides, a senior ultrasonographer performed a BI-RADS assessment in categories2 to5. Additionally, the ultrasonographer manually outlined the breast lesions to obtain ground truth segmentations. Furthermore, 5- and 10-fold cross-validation partitions are provided to standardize the training and test sets to evaluate and reproduce CAD systems. Finally, to validate the utility of the BUS dataset, an evaluation framework is implemented to assess the performance of deep neural networks for segmenting and classifying breastlesions. The BUS dataset is publicly available for academic and research purposes through an open-access repository under the name BUS-BRA: A Breast Ultrasound Dataset for Assessing CAD Systems. BUS images and reference segmentations are saved in Portable Network Graphic (PNG) format files, and the dataset information is stored in separate Comma-Separated Value (CSV)files. The BUS-BRA dataset can be used to develop and assess artificial intelligence-based lesion detection and segmentation methods, and the classification of BUS images into pathological classes and BI-RADS categories. Other potential applications include developing image processing methods like despeckle filtering and contrast enhancement methods to improve image quality and feature engineering for imagedescription.

Complexity and algorithm of setting optimal location for data sink in real-time NOMA-based IIoTs

Real-time performance is one of the most vital metrics in Non-Orthogonal Multiple Access (NOMA) based Industrial Internet of Things (IIoTs) applications. Since the relative geographic relationship between data sink and wireless sensors affects the transmission parallelism and thus the real-time performance, setting a reasonable location for the data sink is thus a feasible way to high real-time performance for applications where the locations of wireless sensors are fixed. In this paper, we consider how to find an optimal location for the data sink to minimize average access delay. We first formulate the problem and prove it to be NP-Complete by presenting an original reduction proof from classic set partition problem. Second, by tightening the NOMA decoding constraint of the original problem, a heuristic algorithm, which is designed based on Apollonian Circle Theorem and Dilworth Theorem, obtains a feasible location for the data sink. Simulation results reveal that the real-time performance loss of the heuristic algorithm is no more than 50% of the best one. Compared with the classic TDMA scheme, the real-time performance increases by more than 60% for some typical settings, and it can even reach 70% for the linear network topology, owing to the full exploitation of NOMA parallelism.

Multi-Intensity Planes Constellation and Code Design-Based Color-Shift Keying for Visible Light Communications

In this study, we develop multi-intensity planes color-shift keying (MIP-CSK) constellations to determine the effect of the MIP-based CSK design scheme on the reliability of visible light communication systems under the constraint of target color constraint. Based on a two-dimensional triangle partition constellation plane, we introduce algorithms to design the non-integral powers of two MIP-CSK constellations in three-dimensional symbol space by using set partition and intensity plane levels, which improves the minimum Euclidean distance. We then develop a coding scheme by designing an finite-state machine from the non-integral powers of two constellations to further improve the reliability under the white color constraint. Subsequently, by considering the spatial symmetric nature of the triangle partition constellation distribution, we study the design strategy of powers of two constellation symbols and further propose the powers of two MIP-CSK design scheme in a three-dimensional symbol space. Moreover, we investigate the reliability benefit obtained by set partition and optimal intensity levels for powers of two MIP-CSK constellations by improving the normalized minimum squared distance. The simulation results indicate that the error performance of the proposed new family of MIP-based design schemes outperforms its counterparts while complying with the white color constraint at the high signal-to-noise ratios.

Minimum Cost Flow-Based Integrated Model for Electric Vehicle and Crew Scheduling

Vehicle and crew scheduling is vital in public transit planning. Conventionally, the issues are handled sequentially as the vehicle scheduling problem (VSP) and crew scheduling problem (CSP). However, integrating these planning steps offers additional flexibility, resulting in improved efficiency compared with sequential planning. Given the ever-growing market share of electric buses, this paper introduces a new model for integrated electric VSP and CSP, called EVCSPM. This model employs the minimum cost flow formulations for electric VSP, set partitioning for CSP, and linking constraints. Due to the nonlinear integer property of EVCSPM, we propose a method that hybrids a matching-based heuristic and integer linear programming solver, GUROBI. The numerical results demonstrate the efficiency of our methodology, and the integrated model outperforms the sequential model in real-life scenarios.

Battery-powered automated guided vehicles scheduling problem in automated container terminals for minimizing energy consumption

Battery-Powered Automated Guided Vehicles (B-AGVs) are important equipment used to transfer containers between the seaside and the landside in automated container terminals. Due to their limited battery capacity, the B-AGVs require battery recovery when their battery reaches the threshold. However, the processing capacity of electrical battery swapping stations is limited, it is crucial to avoid congestion in swapping station. Considering that the scheduling problem of B-AGV in the terminal is time intensive, this paper proposes a set partitioning method based on breadth-first and depth-first search to solve the established mixed integer programming model. Combined with the operational characteristics of the terminal, this paper adopts the flexible pooling strategy to effectively matches the inbound and outbound containers to form dual-cycle operation of B-AGVs, which can reduce the computational complexity and thus improving the solution efficiency. In addition, the record-to-record travel algorithm is applied to improve the quality of the solution. Finally, the effectiveness of the proposed model and solution method is evaluated through numerical experiments. Besides, the applicability of the flexible pooling strategy to different container terminals is studied, and the impact of the number of B-AGVs and the capacity of swapping stations on the energy consumption of B-AGVs is investigated, and some managerial insights are given accordingly.

What is a decision problem?

This paper presents a general framework about what is a decision problem. Our motivation is related to the fact that decision analysis and operational research are structured (as disciplines) around classes of methods, while instead we should first characterise the decision problems our clients present us. For this purpose, we define a decision problem as a set partitioning problem and we introduce a new framework, independent from any existing method, based upon primitives provided by (or elicited from) the client and four fundamental problem statements. We show that constructing the set to partition is a decision problem itself and that the types of primitives are finite. The result is that the number of archetypal decision problems are finite and so the archetypal decision support methods.

Deterministic stack-sorting for set partitions

Deterministic stack-sorting for set partitions

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K*λKv*, the λ-fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K*λKv* has a D-decomposition is called the λ-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ-fold spectrum problem for each of the remaining 21 orientations.

Subgraph of unitary Cayley graph of matrix algebras induced by idempotent matrices

In this work, we are interested in the subgraph of unitary Cayley graph of a matrix algebra over a finite field induced by the set of idempotent matrices. We study its structure and completely obtain all connected components with the degree on each vertex. Most of them turn out to be a regular bipartite graph such that each partite set has the same number of vertices. Our main tools are combinatorial results on matrix algebras. Moreover, we compute the clique number, chromatic number and independence number of the graph.

Multi-route coordination for bus systems in response to road disruptions

Road disruptions frequently occur in practice, resulting in a productivity loss of bus services and widespread passenger delays. These negative impacts are significant, especially when multiple routes are influenced at the same time. In order to mitigate these impacts, this paper proposes a multi-route coordination approach that collaboratively adjusts multiple bus routes and optimizes bus timetables to provide effective alternative bus services for passengers. Three adjusting strategies are adopted for bus routes with varying passenger demand: detouring, short-running, and cancellation. To address the multi-route coordination problem, a column-generation-based two-stage framework is developed. Concretely, a column generation technique is utilized in the first stage to iteratively generate candidate adaptive bus routes and passenger paths. Following that, an integrated integer programming model is built in the second stage to simultaneously determine bus timetables and the combinations of those adaptive bus routes. After realizing the low computational efficiency for large-scale problems, this paper designs a customized decomposition algorithm based on set partitioning to solve the presented model and obtain near-optimal solutions efficiently. Finally, the proposed methodology is applied to an illustrative Sioux Falls network and a real-world bus network in Beijing to verify its validity and effectiveness. Three comparative analyses are conducted to discuss the advantages of the three adjusting strategies, to investigate the benefits of coordinating bus timetables, and to explore the applicability of different adjusting strategies, respectively.

A branch‐and‐price‐based heuristic for the vehicle routing problem with two‐dimensional loading constraints and time windows

AbstractAddressed in this study is a vehicle routing problem with two‐dimensional loading constraints and time windows (2L‐CVRPTW), aiming to minimize the transportation cost while satisfying the two‐dimensional loading and routing constraints with time windows. To solve this problem, for the first time a mixed‐integer linear programming model is formulated with considering practical last‐in‐first‐out loading constraints, and a branch‐and‐price‐based (BP‐based) heuristic is proposed based on a set partitioning formulation. In the heuristic, a modified labeling algorithm is proposed for the complex pricing problem, which is a relaxation of the elementary shortest path problem with resource constraints and two‐dimensional loading constraints. Therein, an effective Tabu‐maximum open space packing heuristic is proposed to verify the feasibility of the two‐dimensional packing problem of each route generated by the labeling algorithm. In addition, effective accelerating and branching strategies are introduced to improve the solving efficiency of the heuristic. To evaluate the effectiveness and the advantages of the proposed heuristic, extensive computational experiments are performed based on the generated instances. The computational results show that the proposed BP‐based heuristic can effectively solve the 2L‐CVRPTW, in which the optimal solutions can be achieved much faster than CPLEX in small‐scale problems. Relationships between the transportation cost and the characteristics of the instances are analyzed. The stability of the algorithm and the effectiveness of the accelerating strategies are verified and discussed.

Local Partition Dimension of Grid Graph and Its Application to the Coordinates of Potential Disaster Areas in Jember Regency

Partition dimension was introduced as a part of interesting topic in graph theory. It was focus to observe about distance. The local partition dimension is an expansion of the partition dimension by adding certain conditions to the representation of the vertices 𝑣 to which is Π denoted by 𝑟(𝑣| Π). One of the conditions that must be met for 𝑟(𝑣| Π) is discussed. The minimum value of k so that there is a local distinguishing partition from V (G) is the local partition dimension of G or it can can be said that the distance of each neighbor is different. The local partition dimension of a graph G is denoted 〖pd〗_l (G). In this study, we used an axiomatic deductive methods and pattern recognition. In order to construct the discriminating set on the metric dimension (dim) and the discriminating partition on the partition dimension (pd), the pattern detection method looks for patterns in which the coordinate values are minimum and different. By all observations, the local partition dimensions of P_n×P_m Grid graph has two condition about the results of resolving partition. The Result of local partition dimension of a Grid graph 〖〖pd〗_l (P〗_n×P_m)=2, where n≥2 dan m≥2. In addition, it will decide how to convert the coordinates of areas in the Jember district that are prone to flooding and landslides into metric dimensions. It was about Coordinates of Flood and Landslide Disaster Locations in Jember Regency. The number of local and minimal partition sets generated for flood-prone areas in Jember Regency is 〖pd〗_l (G_Jember)=3.

A characterization of unit interval bigraphs of open and closed intervals

Interval graphs are the intersection graphs of intervals on the real line. Unit interval graphs are interval graphs representable by intervals of unit length. Rautenbach and Szwarcfiter showed that the class of intersection graphs of the open and closed real intervals of unit length is a strict superclass of the class of unit interval graphs. They also characterized this class of graphs.An interval bigraph is a bipartite graph where to each vertex we can assign an interval so that vertices in the opposite partite sets are adjacent if and only if their intervals intersect. Analogously we show that the class of finite intersection bigraphs of open and closed intervals of unit length is a strict superclass of the class of unit interval bigraphs. We also characterize this class of bigraphs in terms of forbidden induced subgraphs and a suitable extension of proper interval bigraphs.

Partitioning a 2-edge-coloured graph of minimum degree [formula omitted] into three monochromatic cycles

Lehel conjectured in the 1970s that the vertices of every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph G does not have to be complete. It suffices to require that G has minimum degree at least 3n/4, where n is the order of G, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is tight.Here we continue this line of research, by proving that for every red and blue edge-colouring of an n-vertex graph of minimum degree at least 2n/3+o(n), there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

Multiscale mobility patterns and the restriction of human movement.

From the perspective of human mobility, the COVID-19 pandemic constituted a natural experiment of enormous reach in space and time. Here, we analyse the inherent multiple scales of human mobility using Facebook Movement maps collected before and during the first UK lockdown. Firstly, we obtain the pre-lockdown UK mobility graph and employ multiscale community detection to extract, in an unsupervised manner, a set of robust partitions into flow communities at different levels of coarseness. The partitions so obtained capture intrinsic mobility scales with better coverage than nomenclature of territorial units for statistics (NUTS) regions, which suffer from mismatches between human mobility and administrative divisions. Furthermore, the flow communities in the fine-scale partition not only match well the UK travel to work areas but also capture mobility patterns beyond commuting to work. We also examine the evolution of mobility under lockdown and show that mobility first reverted towards fine-scale flow communities already found in the pre-lockdown data, and then expanded back towards coarser flow communities as restrictions were lifted. The improved coverage induced by lockdown is well captured by a linear decay shock model, which allows us to quantify regional differences in both the strength of the effect and the recovery time from the lockdown shock.

Optimizing aircrew recovery considering long connections: A column generation based approach

Severe weather and air traffic flow control can often cause flight delays and disrupt the crew schedules, resulting in a too-short or a too-long connection between flights. Short connections need to be adjusted to ensure the airline’s smooth operations, while the long connection whose duration is only a bit lower than the threshold enabling the crew to leave the flying zone for a rest at airport terminals can be quite inefficient, as the crew’s ground standby is viewed as working all the time without executing any flight which adds to the shortage of crew resources. Thus, we propose a method of delaying flights to the point enabling crews to leave for a rest, which helps save the available working hours. The difficulty lies in the trade-off between the additional swap or delay cost in creating long connections and the possible global cost decrease due to savings in available working hours brought by long connections, which calls for optimization. In this paper, we introduce the crew recovery problem for one day of operation taking long connections into account. The problem is mathematically formulated as an arc-based integer programming model and a set partitioning model in order to determine the optimal flight recovery schedule and the crew reassignment. A column generation based solution approach is developed and the sub-problems consisting of the resource-constrained shortest path problem are solved by a labeling algorithm. Computational experiments based on real-world data from a major Chinese airline confirm the effectiveness and efficiency of the solution approaches. It is also demonstrated the save of available working hours resulting from adaptively creating long connections would decrease the recovery cost by about 18.87%–74.88%.

Hamilton cycles passing through a matching in a bipartite graph with high degree sum

Let G be a balanced bipartite graph of order 2n with partite sets X and Y, and M be a matching of k edges in G. Let σ1,1(G)=min⁡{dG(x)+dG(y):x∈X,y∈Y,xy∉E(G)}. Conditions on σ1,1(G) for the existence of a Hamilton cycle passing through M have been studied by many researchers, but the threshold is not known to be sharp for 2n3≤k≤n−2.In this paper we prove that G has a Hamilton cycle passing through all the edges of M if σ1,1(G)≥2n−k and k≤n−2. Moreover, we show that the lower bound on σ1,1(G) is sharp for every k with 2n3≤k≤n−2.

Orthogonal cycle systems with cycle length less than 10

AbstractAn ‐decomposition of a graph is a partition of the edge set of into subsets, where each subset induces a copy of the graph . A ‐orthogonal ‐decomposition of is a set of ‐decompositions of such that any two copies of in distinct ‐decompositions intersect in at most one edge. When , we call the ‐decomposition an ‐system of order . In this paper, we consider the case is an ‐cycle and construct a pair of orthogonal ‐cycle systems for all admissible orders when , except when .

Cumulants, spreadability and the Campbell–Baker–Hausdorff series

We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and Möbius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faà di Bruno’s theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell–Baker–Hausdorff series in this context. Moreover, we exhibit an interpretation of the Campbell–Baker–Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients.