Partitioning Problem Research Articles

Optimal Network Partition and Edge Server Placement for Distributed State Estimation

This paper investigates network partition and edge server placement problem to exploit the benefit of edge computing for distributed state estimation. A constrained many-objective optimization problem is formulated to minimize the cost of edge server deployment, operation, and maintenance, avoid the difference in the partition sizes, reduce the level of coupling between connected partitions, and maximize the inner cohesion of each partition. Capacities of edge server are constrained against underload and overload. To efficiently solve the problem, an improved non-dominated sorting genetic algorithm III (NSGA-III) is developed, with a specifically designed directed mutation operator based on topological characteristics of the partitions to accelerate convergence. Case study validates that the proposed formulations effectively characterize the practical concerns and reveal their trade-offs, and the improved algorithm outperforms existing representative ones for large-scale networks in converging to a near-optimal solution. The optimized result contributes significantly to real-time distributed state estimation.

A Hybrid Simulated Annealing Algorithm for Solving Continuous Quadratic Programming for the Discrete Balanced Graph Partitioning Problem

A Hybrid Simulated Annealing Algorithm for Solving Continuous Quadratic Programming for the Discrete Balanced Graph Partitioning Problem

State Space Partitioning Based on Constrained Spectral Clustering for Block Particle Filtering

The particle filter (PF) is a powerful inference tool widely used to estimate the filtering distribution in non-linear and/or non-Gaussian problems. To overcome the curse of dimensionality of PF, the block PF (BPF) inserts a blocking step to partition the state space into several subspaces or blocks of smaller dimension so that the correction and resampling steps can be performed independently on each subspace. Using blocks of small size reduces the variance of the filtering distribution estimate, but in turn the correlation between blocks is broken and a bias is introduced. When the dependence relationships between state variables are unknown, it is not obvious to decide how to split the state space into blocks and a significant error overhead may arise from a poor choice of partitioning. In this paper, we formulate the partitioning problem in the BPF as a clustering problem and we propose a state space partitioning method based on spectral clustering (SC). We design a generalized BPF algorithm that contains two new steps: (i) estimation of the state vector correlation matrix from predicted particles, (ii) SC using this estimate as the similarity matrix to determine an appropriate partition. In addition, a constraint is imposed on the maximal cluster size to prevent SC from providing too large blocks. We show that the proposed method can bring together in the same blocks the most correlated state variables while successfully escaping the curse of dimensionality.

Hyper-Graph Partitioning for a Multi-Agent Reformulation of Large-Scale MILPs

This letter addresses the challenge of solving large-scale Mixed Integer Linear Programs (MILPs). A resolution scheme is proposed for the class of MILPs with a hidden constraint-coupled multi-agent structure. In particular, we focus on the problem of disclosing such a structure to then apply a computationally efficient decentralized optimization algorithm recently proposed in the literature. The multi-agent reformulation problem consists in manipulating the matrix defining the linear constraints of the MILP so as to put it in a singly-bordered block-angular form, where the blocks define local constraints and decision variables of the agents, whereas the border defines the coupling constraints. We translate the matrix reformulation problem into a hyper-graph partitioning problem and introduce a novel algorithm which accounts for the specific requirements on the singly-bordered block-angular form to best take advantage of the decentralized optimization approach. Numerical results show the effectiveness of the proposed hyper-graph partitioning algorithm.

LNG Distribution Optimization using Set Partitioning Problem Method

The Indonesian government’s commitment to increase the use of gas for domestic demand, by issuing the Decree of the Minister of Energy and Mineral Resources Number 13K/13/MEM/2020 concerning the Assignment of the Implementation of the Supply and Development of LNG Infrastructure and the Gasification 52 power plants in Indonesia. Therefore, a study on the supply chain design that can support the gasification process of the 52 power plants is crucial. Power plant data is imperative to identify receiving terminals which are then grouped into 8 clusters using the K-Means method. The design will use operating LNG refineries, which will then go to the hub for each cluster. Considering the feasibility factor of the receiving terminal and using the center of gravity method carries the determination of the hub. Considering the investment and operational costs form the feasible route as the most optimal by using the Set Partitioning Problem (SPP) method. The optimization considers several types of ships with six different sizes. The total investment cost required was $107,815,749.11, and the operational cost was $68,993,709.11. The results of the economic analysis indicate the distribution will reach a payback period within 10 years if gas sales use a margin of 1,75 USD.

On a Real-World Railway Crew Scheduling Problem

A railway company has to deal with many interrelated planning tasks. In this work we consider the Railway Crew Scheduling Problem (RCSP) for the Rail Cargo Austria (RCA), which is the largest railway company for freight transportation in Austria and one of the largest in Europe. The RCSP aims for determining the most efficient combination of shifts. For RCA’s purposes these are shifts for engine drivers. A shift consists of a sequence of consecutive scheduled trips of locomotives and tasks over a given period of time. Each trip must be covered by exactly one shift while operational, legal and labor constraints are satisfied. The RCSP is known to be difficult to solve and there exists a wide range of solution approaches resulting from many research activities. We present and investigate a matheuristic to tackle the RCSP. We use a breadth-first search construction heuristic and formulate a Set Partitioning Problem with the aim of minimizing the overall paid working time, while all RCA specific constraints are incorporated. Although schedule efficiency and employee satisfaction are in general conflicting, a cost-efficient schedule will not be implemented if it does not reach acceptance of the crew. In a computational study we evaluate the proposed approach on it’s practical applicability on real-world data provided by RCA, based on the Austrian railway network. The focus is on analyzing the effects of different conditions on crew schedules. In the course of this evaluation we make use of algorithms from previous work on the Locomotive Scheduling Problem (LSP), tailored to RCA’s demands. The LSP is concerned with assigning locomotives to a train schedule while costs are minimized, and the obtained optimal locomotive schedule forms the input for the RCSP. This work provides the basis for a succeeding planning tool, as soon as the locomotive schedule was determined.

Risk-Based planning and partitioning of multiple micro-grids in distribution systems

The rapid fluctuations of renewable power generation such as wind turbines and photovoltaic power plants make significant risks in Multiple Micro-Grids (MMGs) planning and partitioning problem. To control the risk, in this paper a risk-based model is presented for optimal planning and partitioning of MMGs under uncertainties. In addition, two term technical risk and economical risk are introduced to analyze the total risk on MMGs. Genetic algorithm is used for solving the optimization problem. Distributed energy resources and storages placement in MGs and tie switches status for MGs connection to grid or MGs neighbors have been determined, simultaneously. A voltage stability index is formulated and applied for the optimal placement of MMGs via AC power flow tool. Simulation studies is established on the 33-bus IEEE test distribution system and the consequent discussions demonstrate the efficiency of the proposed methodology.

Application of optimal set partitioning theory to solving problems of artificial intelligence and pattern recognition

The paper substantiates the possibility of applying the mathematical theory of continuous problems of optimal partitioning of sets of n-dimensional Euclidean space, which belong to the non-classical problems of infinite-dimensional mathematical programming, to the solution of problems of artificial intelligence and pattern recognition. The problems of pattern recognition both in conditions of certainty and in conditions of uncertainty are formulated. A particular attention is paid to the application of methods of the theory of optimal partitioning for the construction of fuzzy Voronoi diagrams. Examples of constructing fuzzy Voronoi diagrams with the optimal placement of generating points are given.

Fast algorithm of equitably partitioning degenerate graphs into graphs with lower degeneracy

A q-degenerate k-partition of a graph G is a collection (V1,V2,…,Vk) of k pairwise disjoint subsets of V(G) such that V(G)=⋃i=1kVi and each Vi induces a q-degenerate subgraph. Such a partition is called equitable if ||Vi|−|Vj||≤1 for every 1≤i<j≤k. Equitable partition of graphs can model the problem of partitioning a large network into smaller sub-modules based on some cardinal principles, and has many other applications in network science and information science. In this work, we establish theoretical and algorithmic results on equitably partitioning degenerate graphs into graphs with lower degeneracy. Specifically, we show that every n-vertex d-degenerate graph with maximum degree at most n/β has a q-degenerate k-partition (V1,V2,…,Vk) for every k≥αd so that each Vi has size at most ⌈n/k⌉ whenever q,α, and β satisfy a well-defined inequality, and such a partition can be computed in cubic time.

High quality hypergraph partitioning for logic emulation

Dividing a large-scale design into parts for hardware prototyping and emulation is very important for integrated circuit design flow. In this paper, we present a suite of algorithms to solve the partitioning problem in general field programmable gate arrays (FPGAs) prototyping and hardware emulation. Our k-way hypergraph partitioner is based on a multi-level framework and its performance is similar to the well-known partitioners hMetis (Karypis et al., 1999) and KaHyPar (Akhremtsev et al., 2017). Based on this framework, we take up some algorithms which solve partitioning problems with varying objectives and constraints extracted from logic emulation scenarios. These objectives and constraints take FPGA resources, interconnection resources, flip-flops, and clock domains into account, which are all important factors in FPGA prototyping and hardware emulation. We test our algorithms using ISPD98 benchmarks and several industry cases. The experimental results show that our algorithms are efficient and effective in solving different partitioning problems.

Translational surface coupling along a line with non-conforming meshes

In the modelling of large-scale metal structures, comprising plated components intersecting along weld lines, the meshing efficiency and flexibility are limited by the requirements of nodal alignment and compliance of element size, shape and edge orientation throughout the domain. Such limitations necessitate the use of complex transitional meshes in intersection regions and result in highly complex global mesh configurations. This paper presents an original and systematic methodology for surface coupling along an arbitrary 1D interface, which is applicable to any type of 2D and 3D FEs, and which provides a systematic framework for: (i) geometric modelling of weld lines; (ii) coupling of regions with different levels of discretisation detail or element types within a system; and (iii) domain partitioning problems involving computationally heterogeneous partitions. The strategy is based upon a novel coupling element formulation, which uses the fundamental principles of the mortar method and an augmented Lagrangian Multiplier optimisation approach. Particular consideration is given to an element formulation that enforces rigid translational coupling, which has been implemented for employment with co-rotational Reissner-Mindlin shell elements. Various numerical examples are presented to demonstrate the accuracy, versatility and substantial computational benefits of the developed methodology for modelling large-scale metal structural systems.

K-way Balanced Graph Partitioning for Parallel Computing

In modern computing, high-performance computing (HPC) and parallel computing require most of the decision-making in terms of distributing the payloads (input) uniformly across the available set of resources, majorly processors; the former deals with the hardware and its better utilization. In parallel computing, a larger, complex problem is broken down into multiple smaller calculations and executed simultaneously on several processors. The efficient use of resources (processors) plays a vital role in achieving the maximum throughput which necessitates uniform load distribution across available processors, i.e. load balancing. The load balancing in parallel computing is modeled as a graph partitioning problem. In the graph partitioning problem, the weighted nodes represent the computing cost at each node, and the weighted edges represent the communication cost between the connected nodes. The goal is to partition the graph G into k partitions such that: I) the sum of weights on the nodes is approximately equal for each partition, and, II) the sum of weights on the edges across different partitions is minimum. In this paper, a novel node-weighted and edge-weighted k-way balanced graph partitioning (NWEWBGP) algorithm of O(n x n) is proposed. The algorithm works for all relevant values of k, meets or improves on earlier algorithms in terms of balanced partitioning and lowest edge-cut. For evaluation and validation, the outcome is compared with the ground truth benchmarks.

Multi-manned assembly line balancing problem with dependent task times: a heuristic based on solving a partition problem with constraints

This paper aims to study a variant of the multi-manned assembly line balancing problem (MALBP), which considers the possibility of multiple workers simultaneously performing different tasks at the same workstation. In most cases it is assumed that task times are deterministic. This paper takes into account possible interferences between workers and deals with the MALBP with task times depending on the number of workers at the station. Different procedures are developed: resolutions on the basis of a mathematical model, two Relax-and-Fix procedures, a heuristic based on solving a partition problem with constraints (named “HEUR_PART”) and a set of other variants of the HEUR_PART procedure. The computational experiments indicate that HEUR_PART and the HEUR_PART_SGL variant are the proposals that perform best. Additionally, we show that they obtain better results than the ones published in the literature.

A column generation-based exact solution method for seru scheduling problems

This paper attempts to address production scheduling problems in seru production systems (SPS), which is a new-type manufacturing system emanating from Japanese electronic assembly industry. As a typical parallel production system, SPS has high efficiency, good flexibility, and rapid responsiveness, which are achieved by reconfiguring serus, increasing, or decreasing workers to adjust the output in practical volatile markets. The seru scheduling problem in this paper is formulated as an integer programming (IP) model to minimize the total weighted completion time (TWCT). Then, by employing the Dantzig-Wolfe decomposition, the proposed IP model is reformulated into a set partitioning problem with a master problem and several subproblems. A branch-and-bound (B&B) algorithm is designed, and each node in a B&B tree is a linear relaxation problem (LRP) of the set partitioning problem. The LRP is solved by a column generation approach, in which each column is generated to represent a schedule of the seru in SPS based on solving the subproblems. Computational experiments are conducted, and the results indicate that the proposed column generation-based exact solution method is promising in solving the seru scheduling problem effectively.

On the Packing Partitioning Problem on Directed Graphs

This work is aimed to continue studying the packing sets of digraphs via the perspective of partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions for a given digraph D, called the packing partition number of D. Some lower and upper bounds on this parameter are proven, and their exact values for directed trees are given in this paper. In the case of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of minimum cardinality. We also consider this parameter in digraph products. In particular, a complete solution to this case is presented when dealing with the rooted products.

More Bisections by Hyperplane Arrangements

A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the "non-negative side" of the arrangement $\{x\in R^d : p_{H}(x)\ge 0\}$ is the same as the measure on the "non-positive" side $\{x\in R^d : p_{H}(x)\le 0\}$. In 2017 Barba, Pilz \& Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $d\ge \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\ge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevi\'c, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Gr\"unbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $R^{2^a+\ell}$, where $1\leq \ell\leq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.

A Modified Binary Firefly Algorithm to Solve Hardware/Software Partitioning Problem

Hardware/Software (Hw/Sw) partitioning is a crucial step in Hw/Sw co-design that determines which components of the embedded system could be implemented on hardware and which ones on software. It aims to find a design implementation that fulfils all the specification requirements (functionality, goals and constraints) with a low cost. Most formulations of Hw/Sw partitioning dilemma have proven to NP-hard optimization problem. The firefly algorithm (FA) emerges as a significant tool of Swarm Intelligence that has been applied in many areas of optimization. The main purpose of this paper is to present a modified binary firefly algorithm to solve Hw/Sw partitioning problems. Results are compared with other well-known metaheuristic algorithms: Binary Differential Evolution (BDE) , Discrete Differential Evolution(DDE), Self-adaptive differential evolution algorithm (saBDE), New Modified Binary Differential Evolution (NMBDE), New Binary Differential Evolution(NBDE) , Particle Swarm Optimization Algorithm (PSO), and Genetic Algorithm (GA). The computational results show that it produced better results than the all used algorithms.

Incremental partitioning for efficient spatial data analytics

Big spatial data has become ubiquitous, from mobile applications to satellite data. In most of these applications, data is continuously growing to huge volumes. Existing systems for big spatial data organize records at either the record-level or block-level. Systems that use record-level structures include key-value stores and LSM-Tree stores, which support insert and delete operations and they are optimized for highly-selective queries. On the other hand, systems like GeoSpark that use block-level structures (e.g. 128 MB each) are more efficient for analytical queries, but they cannot incrementally maintain the partitioned data and do not support delete operations. This paper proposes a general framework that enables block-level systems to incrementally maintain spatial partitions, in the presence of bulk insertions and deletions, in distributed file system (DFS) blocks. We first formally study the incremental spatial partitioning problem for big data and demonstrate its NP-hardness. Then, we propose a cost model to estimate the performance of queries on the partitioned data and the effect of modifying it as the data grows. After that, we provide three different implementations of the incremental partitioning framework. Comprehensive experiments on large real datasets show that our proposed partitioning algorithms outperforms state-of-the-art spatial partitioning methods.

On certain partition bijections related to Euler's partition problem

We give short elementary expositions of combinatorial proofs of some variants of Euler's partition identity that were first addressed analytically by George Andrews, and later combinatorially by others. The method using certain matrices to concisely explain these bijections, based on ideas first used in a previous manuscript by the author, enables us to also give new generalizations of two of these results.

Coherent network partitions: Characterizations with cographs and prime graphs

We continue to study coherent partitions of graphs whereby the vertex set is partitioned into subsets that induce biclique spanned subgraphs. The problem of identifying the minimum number of edges to obtain biclique spanned connected components (CNP), called the coherence number, is NP-hard even on bipartite graphs. Here, we propose a graph transformation geared towards obtaining an O(log⁡n)-approximation algorithm for the CNP on a bipartite graph with n vertices. The transformation is inspired by a new characterization of biclique spanned subgraphs. In addition, we study coherent partitions on prime graphs, and show that finding coherent partitions reduces to the problem of finding coherent partitions in a prime graph. Therefore, these results provide future directions for approximation algorithms for the coherence number of a given graph.