Set Partitioning Research Articles

The cluster structure function.

For each partition of a data set into a given number of parts there is a partition such that every part is as much as possible a good model (an "algorithmic sufficient statistic") for the data in that part. Since this can be done for every number between one and the number of data, the result is a function, the cluster structure function. It maps the number of parts of a partition to values related to the deficiencies of being good models by the parts. Such a function starts with a value at least zero for no partition of the data set and descents to zero for the partition of the data set into singleton parts. The optimal clustering is the one selected by analyzing the cluster structure function. The theory behind the method is expressed in algorithmic information theory (Kolmogorov complexity). In practice the Kolmogorov complexities involved are approximated by a concrete compressor. We give examples using real data sets: the MNIST handwritten digits and the segmentation of real cells as used in stem cell research.

Topological structures induced by chromatic partitioning of vertex set of graphs

This paper presents a method of constructing topologies on vertex set of a graph G induced by chromatic partition of vertex set of the graph. It introduces colour lower approximation and colour upper approximation of vertex induced subgraphs and acquaints the open and closed sets of the topology generated by chromatic partition on the vertex set of graphs. It explores some of the properties of colour lower approximation and colour upper approximation of vertex induced subgraphs. It also establishes some new subgraphs based on the colour lower approximation and colour upper approximation and some of their properties have been studied.

Optimal design of line replaceable units

AbstractA line replaceable unit (LRU) is a collection of connected parts in a system that is replaced when any part of the LRU fails. Companies use LRUs as a mechanism to reduce downtime of systems following a failure. The design of LRUs determines how fast a replacement is performed, so a smart design reduces replacement and downtime cost. A firm must purchase/repair a LRU upon failure, and large LRUs are more expensive to purchase/repair. Hence, a firm seeks to design LRUs such that the average costs per time unit are minimized. We formalize this problem in a new model that captures how parts in a system are connected, and how they are disassembled from the system. Our model optimizes the design of LRUs such that the replacement (and downtime) costs and LRU purchase/repair costs are minimized. We present a set partitioning formulation for which we prove a rare result: the optimal solution is integer, despite a nonintegral feasible polyhedron. Second, we formulate our problem as a binary linear program (BLP). The article concludes by numerically comparing the computation times of both formulations and illustrates the effects of various parameters on the model's outcome.

An Enhanced Compression Method for Medical Images Using SPIHT Encoder for Fog Computing

When it comes to filtering and compressing data before sending it to a cloud server, fog computing is a rummage sale. Fog computing enables an alternate method to reduce the complexity of medical image processing and steadily improve its dependability. Medical images are produced by imaging processing modalities using X-rays, computed tomography (CT) scans, magnetic resonance imaging (MRI) scans, and ultrasound (US). These medical images are large and have a huge amount of storage. This problem is being solved by making use of compression. In this area, lots of work is done. However, before adding more techniques to Fog, getting a high compression ratio (CR) in a shorter time is required, therefore consuming less network traffic. Le Gall5/3 integer wavelet transform (IWT) and a set partitioning in hierarchical trees (SPIHT) encoder were used in this study’s implementation of an image compression technique. MRI is used in the experiments. The suggested technique uses a modified CR and less compression time (CT) to compress the medical image. The proposed approach results in an average CR of 84.8895%. A 40.92% peak signal-to-noise ratio (PSNR) PNSR value is present. Using the Huffman coding, the proposed approach reduces the CT by 36.7434 s compared to the IWT. Regarding CR, the suggested technique outperforms IWTs with Huffman coding by 12%. The current approach has a 72.36% CR. The suggested work’s shortcoming is that the high CR caused a decline in the quality of the medical images. PSNR values can be raised, and more effort can be made to compress colored medical images and 3-dimensional medical images.

The k-Ramsey number of two five cycles

Given any two graphs F and H, the Ramsey number R(F, H) is defined as the smallest positive integer n such that every red-blue coloring of the edges of the complete graph Kn of order n, there will be a subgraph of Kn isomorphic to F whose edges are all colored red (a red F) or a subgraph of Kn isomorphic to H whose edges are all colored blue (a blue H). If F and H are bipartite graphs, then the k-Ramsey number R k ( F , H ) is defined as the smallest positive integer n such that for any red-blue coloring of the edges of the complete k-partite graph of order n in which each partite set is of order ⌊ n k ⌋ or ⌈ n k ⌉ there will be a subgraph isomorphic to F whose edges are all colored red (a red F) or a subgraph isomorphic to H whose edges are all colored blue (a blue H). Andrews, Chartrand, Lumduanhom and Zhang found the k-Ramsey number R k ( F , H ) for F = H = C 4 , and for F = K 1 , t and H = K 1 , s where s , t ≥ 2 . We continue their work by investigating the case where the graphs F and H are both C 5.

An Empirical Selection of Wavelet for Near-Lossless Medical Image Compression.

Wavelets are defined as mathematical functions that segment the data into different frequency levels. We can easily capture the fine and coarse details of an image or signal referred to as a subband. And it also helps in subband thresholding to achieve good compression performance. In recent days in telemedicine services, the handling of medical images is prominently increasing and it leads to the demand for medical image compression. While compressing the medical images, we have to concentrate on the data that holds important information, and at the same time, it must retain the image quality. Near-Lossless compression plays an essential role to achieve a better compression ratio than lossy compression and provides better quality than lossless compression. In this paper, we analyzed the sub-banding of Discrete Wavelet Transform (DWT) using different types of wavelets and made an optimal selection of wavelets for subband thresholding to attain a good compression performance with an application to medical images. We used Set Partitioning In Hierarchical Trees (SPIHT) compression scheme to test the compression performance of different wavelets. The Peak Signal to Noise Ratio (PSNR), Bits Per Pixel (BPP), Compression Ratio, and percentage of number of zeros are used as metrics to assess the performance of all the selected wavelets. And to find out its efficiency in possessing the essential information of medical images, the subband of the selected wavelets is further utilized to devise the near-lossless compression scheme for medical images.

Mutual witness Gabriel drawings of complete bipartite graphs

Let Γ be a straight-line drawing of a graph and let u and v be two vertices of Γ. The Gabriel disk of u,v is the disk having u and v as antipodal points. A pair 〈Γ0,Γ1〉 of vertex-disjoint straight-line drawings forms a mutual witness Gabriel drawing when, for i=0,1, any two vertices u and v of Γi are adjacent if and only if their Gabriel disk does not contain any vertex of Γ1−i. We characterize the pairs 〈G0,G1〉 of complete bipartite graphs that admit a mutual witness Gabriel drawing. The characterization leads to a linear time testing algorithm. We also show that when the pair 〈G0,G1〉 consists of two complete multi-partite graphs whose partition sets all have size greater than one, then the pair does not admit a mutual witness Gabriel drawing unless the pair is 〈K2,2,K2,2〉.

The perfect matching and tight Hamilton cycle decomposition of complete n-balanced mk-partite k-uniform hypergraphs

Let r≥k≥2 and Kr×n(k) denote the complete n-balanced r-partite k-uniform hypergraph, whose vertex set consists of r parts, each part has n vertices, and whose edge set contains all the k-element subsets with no two vertices from one part. A Hamilton cycle decomposition of Kr×n(k) is a partition of the edge set E(Kr×n(k)) into Hamilton cycles. In this paper, we consider the perfect matching decomposition and the tight Hamilton cycle decomposition of Kmk×n(k) for m≥2. We obtain the following results. (1) Let k≥3, m≥1 and n≥1. Then Kmk×n(k) has a perfect matching decomposition. (2) If Kmk(k) has a Hamilton cycle decomposition, then Kmk×n(k) has a tight Hamilton cycle decomposition.

Information and helix mechanism of entropy increase

The principle of entropy increase not only constitutes the basis of statistical mechanics but is also closely related to the irreversibility of time, the origin of life, chaos, and turbulence. In this paper, by discussing the definition of the dynamic system of entropy from the perspective of symbols and partition of information, the entropy transfer characteristics based on the set partition are proposed. By introducing the hypothesis of limited measurement accuracy into the dynamic system with continuous phase space, two necessary mechanisms for the formation of chaos are obtained, namely, the transfer of entropy from small scale to macro-scale and the dissipation of information in macroscale. Furthermore, the relationship between the increase in local entropy and the Lyapunov exponent of the dynamic system is established. Then the entropy increase and abnormal dissipation mechanism of helical waves in fluid and general physical systems are discussed.

Lower bounds and properties for the average number of colors in the non-equivalent colorings of a graph

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. We study the average number A ( G ) of colors in the non-equivalent colorings of a graph G . We conjecture several lower bounds on A ( G ) , determine the value of this graph invariant for some classes of graphs and give general properties of A ( G ) which we will use for proving the validity of the conjectures for specific families of graphs, namely chordal graphs and graphs with maximum degree at most 2.

On connectivity of the semi-splitting block graph of a graph

Abstract A graph G is said to be a semi-splitting block graph if there exists a graph H such that SB(H) ≌ G. This paper establishes a characterisation of semi-splitting block graphs based on the partition of the vertex set of G. The vertex (edge) connectivity and p-connectedness (p-edge connectedness) of SB(G) are examined. For all integers a, b with 1 < a < b, the existence of the graph G for which κ (G) = a, κ (SB(G)) = b and λ (G) = a, λ (SB(G)) = b are proved independently. The characterization of graphs with κ(SB(G)) = κ (G) and a necessary condition for graphs with κ (SB(G)) = λ (SB(G)) are achieved.

A review of feature set partitioning methods for multi-view ensemble learning

Since the present era is entirely computer and Internet of Things (IoT) oriented, enormous amounts of data are produced quickly from many sources. Machine learning’s primary function, which is widely applicable to these high-dimensional datasets, is classification. One of the well-known approaches for improving classification effectiveness is ensemble learning. The splitting of huge datasets vertically based on the feature set partitioning (FSP) method is known as views. Multi-view ensemble learning (MEL) is the process of applying different classification models to these views and then combining their predictions. The FSP methods used to create views are the backbone of MEL. This review has been done to study, present, analyze, and collectively compare various FSP methods. Several issues and challenges related to FSP and MEL are also highlighted in this paper. The classification performance of recent FSP methods for the MEL framework is compared as an auxiliary experiment using fifteen high-dimensional datasets. A nonparametric statistical analysis is used for the comparison of FSP methods.

A branch-and-price algorithm for the airport gate assignment problem considering the trade-off between robustness and efficiency

In this paper, a novel objective for the airport gate assignment problem which considers the trade-off between robustness and operational efficiency in a unified dimension is developed. A task-based two-phase model is established, where the branch-and-price framework is embedded in the first stage. A set partitioning model is developed, where a series of aircrafts assigned to the same gate of a certain type are defined as a gate plan. The pricing subproblem is also decomposed into three parallel subproblems according to the gate size to account for the heterogeneity of the gates. Each gate plan is allocated to a physical gate in order to minimize the utilization of remote gates in the second stage. The pulse algorithm with two acceleration strategies, that generates a lower bound matrix in reverse order and applies path combination, is used to solve the pricing subproblems. Finally, extensive computational experiments are conducted to demonstrate the high performance of the proposed model, especially for large-scale instances.

Tolerant Radon partitions on the all-paths convexity in graphs

In a graph G, the all-paths transit function A(u, v), consists of the set of all vertices in the graph G which lies in some path connecting u and v. Convexity obtained by the all-paths transit function is called all-paths convexity. A partition of a set P of vertices of a graph G into two disjoint non-empty subsets is called a Radon partition if the convex hulls of the subsets intersect. A Radon partition (P0, Q0) of P into nonempty subsets is called t-tolerant Radon partition, if for any set S ⊆ P with | S | ≤ t , the intersection of the convex hulls 〈 P 0 − S 〉 ∩ 〈 Q 0 − S 〉 ≠ ∅ . Here we study the t-tolerant Radon number of G with respect to all-paths convexity. It is the minimum number k such that any collection of k vertices of G has t-tolerant Radon partition. We prove that if G has only one block, then k = 2 t + 2 whenever t ≥ 1 . Finally, we prove that if the block-cut vertex tree representation of G is a path then k = 2 t + 3 , and if the block-cut vertex tree representation of G is a tree that is not a path then k = 2 t + 4 .

Characterizing the supernorm partition statistic

The paper gives characterizations of the supernorm statistic widehat{N} :mathcal {P}rightarrow mathbb {N}^{+} for partitions, where mathcal {P} is the set of integer partitions and mathbb {N}^{+} is the positive integers. The supernorm statistic map is a bijection onto mathbb {N}^{+} that defines a total order on partitions, the supernorm ordering, obtained by pulling back the additive total order on mathbb {N}^{+}. The supernorm ordering refines two partial orders on partitions: the multiset inclusion order, whose image under widehat{N} is the divisibility lattice on mathbb {N}^{+}, and the Young’s lattice order. The paper shows that it is characterized by these two properties, additionally with the requirement of an order-isomorphism from the multiset inclusion order to the divisibility order. It is also characterized by these two properties, additionally with the requirement of mapping the partitions with exactly one part bijectively to the prime numbers. It presents a construction showing that the latter additional conditions are necessary for the characterization to hold.

On the existence of k $k$‐cycle semiframes for even k $k$

AbstractA ‐semiframe of type is a ‐group divisible design of type in which is the vertex set, is the group set, and the set of ‐cycles can be written as a disjoint union where is partitioned into parallel classes on and is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition or for some . In this paper, we almost completely solve the existence of a ‐semiframe of type for all and a ‐semiframe of type for all and .

Generalized spikes with circuits and cocircuits of different cardinalities

We consider matroids with the property that every subset of the ground set of size s is contained in a 2s-element circuit and every subset of size t is contained in a 2t-element cocircuit. We say that such a matroid has the (s,2s,t,2t)-property. A matroid is an (s,t)-spike if there is a partition of the ground set into pairs such that the union of any s pairs is a circuit and the union of any t pairs is a cocircuit. Our main result is that all sufficiently large matroids with the (s,2s,t,2t)-property are (s,t)-spikes, generalizing a 2019 result that proved the case where s=t. We also present some properties of (s,t)-spikes.

Local search versus linear programming to detect monotonicity in simplicial branch and bound

AbstractThis study focuses on exhaustive global optimization algorithms over a simplicial feasible set with simplicial partition sets. Bounds on the objective function value and its partial derivative are based on interval automatic differentiation over the interval hull of a simplex. A monotonicity test may be used to decide to either reject a simplicial partition set or to reduce its simplicial dimension to a relative border (at the boundary of the feasible set) facet (or face) by removing one (or more) vertices. A monotonicity test is more complicated for a simplicial sub-set than for a box, because its orientation does not coincide with the components of the gradient. However, one can focus on directional derivatives (DD). In a previous study, we focused on either basic directions, such as centroid to vertex or vertex to vertex directions, or finding the best directional derivative by solving an LP or MIP. The research question of this paper refers to using local search (LS) based sampling of directions from vertex to facet. Results show that most of the monotonic DD found by LP are also found by LS, but with much less computational cost. Notice that finding a monotone direction does not require to find the direction in which a derivative bound is the steepest.

Coding Application

Key words: message code, coding equivalence, subset code, discrete optimization problem, set partitioning, knapsack problem, variant observation algorithm The work is devoted to the development of the algorithm for observing all subsets of the set. For that purpose, coding for subsets of the set is considered, in such a way as to create a 1-1 correspondence between the set of subsets of the set and the set of codes. A convenient arrangement of the observation of all those codes allows the observation of all subsets of the set in a convenient way. Subsets of a set are coded in the binary number system (with binary numbers). The problem of considering subsets of a set arises when solving the knapsack problem, set partitioning problems, and other problems. To the mentioned general problems are included a number of specific problems arising in many fields. In the work, an algorithm for observing all subsets of the set is developed, which is suitable for solving all the mentioned problems. The developed algorithm can be applied to the optimal solution of non-large NP-complete problems. All the known optimal solution algorithms for solving large-scale problems run in practically unacceptable time (have very long execution time). As a consequence, for the solution of large NP-complete problems, approximate algorithms with polynomial time complexity are developed, by means of which approximate solutions of the problem are obtained.

PH-CF: A Phased Hybrid Algorithm for Accelerating Subgraph Matching Based on CPU-FPGA Heterogeneous Platform

Nowadays, more and more data are represented and stored by a graph structure, and subgraph matching is a fundamental problem in a variety of scientific machine learning and industrial applications, such as remote sensing image registration, industrial inspection, etc. Due to the NP-hard problem of subgraph matching, the explosive growth of graph data, the disadvantages of high energy consumption, and the high overhead of CPU and GPU platforms, computing subgraph matching is becoming more and more challenging. To alleviate this problem, we propose a phased hybrid algorithm to accelerate the enumeration task of subgraph matching, called <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PH-CF</monospace> , based on the CPU-FPGA heterogeneous platform. This approach can make full use of the pipeline and data flow mechanism, low power consumption, and configurable characteristics of FPGA. First, the matching order of query vertices automatically selects GQL or RI methods according to the sparsity of the data (query) graph. Second, a candidate vertex auxiliary data structure set partitioning method is designed to effectively realize the load balance of multiple computing units at the FPGA and CPU host sides. Third, FPGA's pipeline and data flow mechanism is used to accelerate the enumeration phase of subgraph matching. Experimental results on real-world and synthetic datasets show that the performance of the <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PH-CF</monospace> outperforms the state-of-the-arts. <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PH-CF</monospace> can obtain the average performance improvement of up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$16.07\times$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$38.61\times$</tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$11.46\times$</tex-math></inline-formula> over CFL, CECI, and DP-iso, respectively. Moreover, our approach has good stability and robustness on various datasets.