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Metaheuristics for the Minimum Gap Graph Partitioning Problem
Abstract
The Minimum Gap Graph Partitioning Problem (MGGPP) consists in partitioning a vertex-weighted undirected graph into a given number of connected subgraphs with the minimum difference between the largest and the smallest weight in each subgraph. We propose a two-level Tabu Search algorithm and an Adaptive Large Neighborhood Search algorithm to solve the MGGPP in reasonable time on instances with up to about 23 000 vertices. The quality of the heuristic solutions is assessed comparing them with the solutions of a polynomially solvable combinatorial relaxation.
... A connected component in a graph is a maximal subset of nodes connected by paths. Finding connected components is one of the most important tasks in the field of graph analysis with various applications including pattern recognition [1,2], reachability indexing [3][4][5], graph compression [6][7][8], graph partitioning [9][10][11], random walk [12], etc. Meanwhile, billion to trillion-scale graphs have emerged recently, which are very challenging to handle because of the enormity. ...
... The sketching step, proposed in [34], computes connected components on each chunk of the input graph to reduce the graph size and does a load balancing work (line 2), where a chunk is a subset of edges existing consecutively in storage. In the partitioning step, UniCon partitions the input graph into the number of partitions ρ overlapping subgraphs by iteratively running the unified star-operation UniStar or the optimized version UniStar-opt (lines [3][4][5][6][7][8][9]. If the number of input edges is less than a threshold τ, UniCon runs Rem instead of UniStar to reduce the number of rounds (lines 4-8). ...
... implies that u has no path to another node through node v, and thus UniStar excludes the edge from G ξ(u) in the next round. In Fig 4, for example, blue edges in round r are intact, and the intact edge sets are I 0 = {(4, 2)} and I 1 = {(4, 1), (11,9), (13,9), (17,16)}. Let G(r), G 0 (r), G i (r), and G 0 i ðrÞ be G, G 0 , G i , and G 0 i in round r, respectively. ...
With a cluster of commodity hardware, how can we efficiently find all connected components of an enormous graph containing hundreds of billions of nodes and edges? The problem of finding connected components has been used in various applications such as pattern recognition, reachability indexing, graph compression, graph partitioning, and random walk. Several studies have been proposed to efficiently find connected components in various environments. Most existing single-machine and distributed-memory algorithms are limited in scalability as they have to load all data generated during the process into the main memory; they require expensive machines with vast memory capacities to handle large graphs. Several MapReduce algorithms try to handle large graphs by exploiting distributed storage but fail due to data explosion problems, which is a phenomenon that significantly increases the size of data as the computation proceeds. The latest MapReduce algorithms resolve the problem by proposing two distinguishing star-operations and executing them alternately, while the star-operations still cause massive network traffic as a star-operation is a distributed operation that connects each node to its smallest neighbor. In this paper, we unite the two star-operations into a single operation, namely UniStar, and propose UniCon, a new distributed algorithm for finding connected components in enormous graphs using UniStar. The partition-aware processing of UniStar effectively resolves the data explosion problems. We further optimize UniStar by filtering dispensable edges and exploiting a hybrid data structure. Experimental results with a cluster of 10 cheap machines each of which is equipped with Intel Xeon E3-1220 CPU (4-cores at 3.10GHz), 16GB RAM, and 2 SSDs of 1TB show that UniCon is up to 13 times faster than competitors on real-world graphs. UniCon succeeds in processing a tremendous graph with 129 billion edges, which is up to 4096 times larger than graphs competitors can process.
... Wu et al. [29] introduced a deterministic annealing neural network algorithm that attempts to obtain a high-quality solution by following a path of minimum points of a barrier problem. Recently, Bruglieri and Cordone [30] proposed a two-level tabu search algorithm and an adaptive large neighborhood search algorithm to solve the minimum gap graph partitioning problem on instances of large-size. These methods have been tested on large-scale graphs providing good solutions in a reasonable time. ...
In this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.
... Other extent directions of the proposal, moving away from chemical applications, include the following: cutting a graph into smaller pieces is one of the fundamental algorithmic operations; partitioning large graphs is often an important sub-problem for complexity reduction or parallelization [113]; the balanced connected k-partition problem is addressed, with (or without [114]) nonnegative weights on vertices [115,116]; in the same context, the minimum gap graph partitioning problem is addressed [117]; partition strategies on resource description framework graphs have been studied [118,119]; graph contraction (creating of a graph minor [120]) serves in some specific graph-related problems [121]; when parallel motif discovery is employed in complex networks [122], graph partitioning divides the network for an efficient parallelization (divide into an approximately equal number of vertices to parts); strategies based on spectral [123] (eigenproblem in [11]), combinatorial [124], geometric [125], or multi-level [126] heuristics are available. ...
Some polynomials find their way into chemical graph theory less often than others. They could provide new ways of understanding the origins of regularities in the chemistry of specific classes of compounds. This study’s objective is to depict the place of polynomials in chemical graph theory. Different approaches and notations are explained and levelled. The mathematical aspects of a series of such polynomials are put into the context of recent research. The directions in which this project was intended to proceed and where it stands right now are presented.
... There is a large number of literature on methods that solve graph partitioning problem, including spectral partitioning (Pothen et al. 1990), geometric partitioning (Hungershöfer and Wierum 2002), streaming graph partitioning (Abbas et al. 2018), intelligent optimization algorithm (Bruglieri and Cordone 2021), linear programing (Nip et al. 2022) and semi-define programing (Lisser and Rendl 2003). ...
Graph partitioning is a classical NP problem. The goal of graphing partition is to have as few cut edges in the graph as possible. Meanwhile, the capacity limit of the shard should be satisfied. In this paper, a model for graph partitioning is proposed. Then the model is converted into a mixed 0-1 linear programming by introducing variables. In order to solve this model, we select some variables to design the vertex relocation model. This work designs a variable selection strategy according to the effect of vertex relocation on the number of local edges. For purpose of implementing graph partitioning on large scale graph, we design an iterative algorithm to solve the model by selecting some variables in each iteration. The algorithm relocates the shard of the vertex according to the solution of the model. In the experiment, the method in this paper is simulated and compared with BLP and its related methods in the different shard sizes on the five social network datasets. The simulation results show that the method of this paper works well. In addition, we compare the effects of different parameter values and variables selection strategies on the partitioning effect.
... Their algorithm was first introduced to partition a graph into two portions. In 2021, Bruglieri and Cordone [18] proposed meta-heuristics to partition undirected graphs with weighted vertices in such a way that the difference between the greatest weight and the least weight in each subgraph are minimized. Their approach comprises a two-level Tabu search besides an adaptive large neighborhood search method. ...
One of the crucial research areas in the analysis of complex social networks is the identification of communities. Since community detection is an NP-complete problem, numerous meta-heuristic approaches have been used for this problem, mostly taking “modularity” as the objective function. However, modularity-based optimization methods suffer from resolution limit. In this paper, a novel community detection algorithm is proposed that aims to optimize a newly introduced fitness function “balanced modularity.” The proposed method, called grey wolf balanced modularity (GWBM), uses fast label propagation algorithm to form the initial population, relatively novel grey wolf algorithm as the main optimizer, and a problem-specific variant of simulated annealing algorithm for local search. The experiments on synthetic and known real-world networks, including Karate, American Football, and Facebook, have shown that GWBM is accurate and comparable with the state-of-the-art community detection methods.
... With (or without [15]) nonnegative weights on vertices, in [16,17] the balanced connected k-partition problem was addressed, which is known to be NP-hard. In the same context the minimum gap graph partitioning problem was formulated, as addressed in [18]. Partition strategies on resource description framework graphs have been studied in [19] and in [20]. ...
A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in the partition is not relevant (the sets are not distinguished by their positions). The two isomers of C28 fullerenes were colored to test the ability of classifiers to generate different partitions and colorings, thereby providing a useful visual tool for scientists working on the functionalization of various highly symmetrical chemical structures.
In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.
We propose the Multi-Constructor CMSA, a Construct, Merge, Solve and Adapt (CMSA) algorithm that employs multiple heuristic procedures, respectively solution constructors, for the Maximum Disjoint Dominating Sets Problem (MDDSP). At every iteration of the search procedure, the solution components built by the constructors are merged into a sub-instance, which is subsequently solved by an exact solver and then adapted to keep only beneficial solution components. In our CMSA the solution constructors are chosen at random according to their relative probabilities, which are adapted during the search, through a mechanism based on reinforcement learning. We test two variants of the new Multi-Constructor CMSA that employ, respectively, two and six solution constructors, on a new set of 3600 problem instances, encompassing random graphs, Watts–Strogatz networks and Barabási-Albert networks, generated through a Hammersley sampling procedure on the instance space. We compare our algorithm against six heuristics from the literature, as well as with the standard version of CMSA. Furthermore, we employ an integer linear programming (ILP) model that is able to achieve a good performance for small, sparse graphs. Overall, the experimental results show that all versions of CMSA outperform by a large margin the previous state-of-the-art and that, among the variants of CMSA, the novel version that combines two constructors provides slightly better results than the other ones, more prominently on larger graphs.
This article deals with the problem of partitioning a graph into connected components by optimizing some balancing objective functions related to the vertex weights. Objective functions based on the gap or range of the partition's components, that is, the difference between the maximum and minimum weight of a vertex in the component, have been already introduced in the literature. Here we introduce the notion of aggregated gap , defined as the sum of the differences between the weights of the vertices and the minimum weight of a vertex in the component. We study new connected ‐partitioning problems whose objective is a function of the components' aggregated gap, and give NP‐hardness results for these problems on general graphs. Mathematical programming formulations are proposed for these problems adopting flow‐based constraints for modeling connectivity in a partition. Even if they are introduced for the new aggregated gap problems, such formulations are rather general and apply also to the classical non‐aggregated gap problems. Extensive computational tests, both for aggregated and non‐aggregated gap problems, are performed on a set of squared grids and randomly generated graphs with up to 120 vertices, and a number of components ranging from 2 to 9. In our experiments, we test several alternative formulations for our problems providing a comparative analysis of their performance.
We propose a “construct, merge, solve and adapt” (CMSA) approach for the maximum disjoint dominating sets problem (MDDSP), which is a complex variant of the classical minimum dominating set problem in undirected graphs. The problem requires to find as many vertex-disjoint dominating sets of a given graph as possible. CMSA is a recent metaheuristic approach based on the idea of problem instance reduction. At each iteration of the algorithm, sub-instances of the original problem instance are solved by an exact solver. These sub-instances are obtained by merging the solution components of probabilistically generated solutions. CMSA is the first metaheuristic proposed for solving the MDDSP. The obtained results show that CMSA outperforms all existing greedy heuristics.
This paper presents the semantic relation-based modularity-optimized community detection algorithm for Heterogeneous Networks. This paper aims to try to increase the modularity value of the network by using the content analysis of the network and the link analysis together. Therefore, similarity values between people's shares were calculated and included in the network as indirect links. The proposed content- and link-based methods are a greedy hierarchical clustering algorithm that uses indirect connections with the network structure and ensures that the nodes most relative to each other are topologically and semantically grouped with priority. This paper presents a comparative analysis to analyze the impact of semantic relation over the optimization algorithm, i.e. Parliamentary Optimization Algorithm (POA) and Modularity Optimization Algorithm (MOA) for community detection. Finally, the modularity and NMI of the proposed work were evaluated over six real network-based heterogeneous network data sets and gained a satisfactory modularity rate over the resultant informative community.KeywordsSocial mediaCommunity detectionClusteringOptimization algorithmModularitySemantic relationHierarchical clustering algorithmSeed community
Graph partitioning is a widely studied problem in the literature with several applications in real life contexts. In this paper we study the problem of partitioning a graph, with weights at its vertices, into p connected components. For each component of the partition we measure the difference between the maximum and the minimum weight of a vertex in the component. We consider two objective functions to minimize, one measuring the maximum of such differences among all the components in the partition, and the other measuring the sum of the differences between the maximum and the minimum weight of a vertex in each component. We focus our analysis on tree graphs and provide polynomial time algorithms for solving these optimization problems on such graphs. In particular, we present an O(n2logn) time algorithm for the min–max version of the problem on general trees and several, more efficient polynomial algorithms for some trees with a special structure, such as spiders and caterpillars. Finally, we present NP-hardness and approximation results on general graphs for both the objective functions.
This paper considers the problem of zone delineation management and crop planning. The problem consists of selecting which crops to plant in different management zones in order to minimize the total costs subjected to a given demand requirement. From a hierarchical point of view, the process starts by generating a partition of an agricultural field into homogeneous management zones, according to a given soil property. Then, the best crop rotation must be assigned to each management zone, applying agronomic practices in a site-specific manner in each zone. This hierarchical approach establishes two decision making levels of planning. At each level, a two-stage stochastic optimization model is proposed, representing the uncertain behavior of a soil property and crop yields by using a finite set of scenarios. Next, we combined them into a new two-stage stochastic program, solving an integrated approach by simultaneously determining an optimal zoning and allocation. Results from a set of evaluated instances showed the relevance of the proposed methodology and the benefits of the hierarchical approach over the integrated one.
In this paper the influence of several factors to establish a suitable DMA design is considered using an adaptation of the methodology previously developed by Gomes [1]. This methodology is based on graph theory concepts (Floyd-Warshall algorithm) and some user-defined criteria to establish the number and size of DMAs and uses a Simulated Annealing algorithm to identify the most appropriate number and location of metering stations, boundary valves and network reinforcement/replacement needs along the project plan. Different design criteria are applied to a case study and the results obtained by the methodology are used to draw some conclusions.
Various multiobjective evolutionary algorithms (MOEAs) have been applied to solve the optimal design problems of a water distribution system (WDS). Such methods are able to find the near-optimal trade-off between cost and performance benefit in a single run. Previously published work used a number of small benchmark networks and/or a few large, real-world networks to test MOEAs on design problems of WDS. A few studies also focused on the comparison of different MOEAs given a limited computational budget. However, no consistent attempt has been made before to investigate and report the best-known approximation of the true Pareto front (PF) for a set of benchmark problems, and thus there is not a single point of reference. This paper applied 5 state-of-the-art MOEAs, with minimum time invested in parameterization (i.e.,using the recommended settings), to 12 design problems collected from the literature. Three different population sizes were implemented for each MOEA with respect to the scale of each problem. The true PFs for small problems and the best-known PFs for the other problems were obtained. Five MOEAs were complementary to each other on various problems, which implies that no one method was completely superior to the others. The nondominated sorting genetic algorithm-II (NSGA-II), with minimum parameters tuning, remains a good choice as it showed generally the best achievements across all the problems. In addition, a small population size can be used for small and medium problems (in terms of the number of decision variables). However, for intermediate and large problems, different sizes and random seeds are recommended to ensure a wider PF. The publicly available best-known PFs obtained from this work are a good starting point for researchers to test new algorithms and methodologies for WDS analysis.
The paper presents a multi-objective approach for the automatic partitioning of a water distribution network into District Metering Areas (DMAs). In the proposed methodology, both technical and economic issues are considered, including reduction of water leakages due to pressure management and energy production achievable through the use of turbines. The optimization approach is based on a clear explanation of the objective functions, whose structures are defined with the aim of providing full control to the decision-maker. Concepts from graph theory and cluster analysis methods are combined in a three-step algorithm, which can be easily implemented for software use.
We present an application of an Adaptive Large Neighborhood Search (ALNS) algorithm to the Resource-constrained Project Scheduling Problem (RCPSP). The ALNS framework was first proposed by Pisinger and Røpke [19] and can be described as a large neighborhood search algorithm with an adaptive layer, where a set of destroy/repair neighborhoods compete to modify the current solution in each iteration of the algorithm. Experiments are performed on the well-known J30, J60 and J120 benchmark instances, which show that the proposed algorithm is competitive and confirms the strength of the ALNS framework previously reported for different variants of the Vehicle Routing Problem.
Heuristics based on large neighborhood search have recently shown outstanding results in solving various transportation and scheduling problems. Large neighborhood search
methods explore a complex neighborhood by use of heuristics. Using large neighborhoods makes it possible to find better candidate
solutions in each iteration and hence traverse a more promising search path. Starting from the large neighborhood search method,
we give an overview of very large scale neighborhood search methods and discuss recent variants and extensions like variable depth search and adaptive large neighborhood search.
We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel
paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem.
An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at
each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to
many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning
problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address
the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph
colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling
evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide
an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying
process and extract some generic guidelines for its future use on other combinatorial problems.
Integer programming has benefited from many innovations in models and methods. Some of the promising directions for elaborating these innovations in the future may be viewed from a framework that links the perspectives of artificial intelligence and operations research. To demonstrate this, four key areas are examined:
1.(1) controlled randomization,
2.(2) learning strategies,
3.(3) induced decomposition and
4.(4) tabu search. Each of these is shown to have characteristics that appear usefully relevant to developments on the horizon.
The pickup and delivery problem with time windows is the problem of serving a number of transportation requests using a limited amount of vehicles. Each request involves moving a number of goods from a pickup location to a delivery location. Our task is to construct routes that visit all locations such that corresponding pickups and deliveries are placed on the same route, and such that a pickup is performed before the corresponding delivery. The routes must also satisfy time window and capacity constraints.
This paper presents a heuristic for the problem based on an extension of the large neighborhood search heuristic previously suggested for solving the vehicle routing problem with time windows. The proposed heuristic is composed of a number of competing subheuristics that are used with a frequency corresponding to their historic performance. This general framework is denoted adaptive large neighborhood search.
The heuristic is tested on more than 350 benchmark instances with up to 500 requests. It is able to improve the best known solutions from the literature for more than 50% of the problems.
The computational experiments indicate that it is advantageous to use several competing subheuristics instead of just one. We believe that the proposed heuristic is very robust and is able to adapt to various instance characteristics.
This paper proposes a construction heuristic and an adaptive large neighborhood search heuristic for the technician and task
scheduling problem arising in a large telecommunications company. This problem was solved within the framework of the 2007
challenge set up by the French Operational Research Society (ROADEF). The paper describes the authors’ entry in the competition
which tied for second place.
We introduce a novel criterion in clustering that seeks clusters with limited range of values associated with each cluster's elements. In clustering or classification the objective is to partition a set of objects into subsets, called clusters or classes, consisting of similar objects so that different clusters are as dissimilar as possible. We propose a number of objective functions that employ the range of the clusters as part of the objective function. Several of the proposed objectives mimic objectives based on sums of similarities. These objective functions are motivated by image segmentation problems, where the diameter, or range of values associated with objects in each cluster, should be small. It is demonstrated that range‐based problems are in general easier, in terms of their complexity, than the analogous similarity‐sum problems. Several of the problems we present could therefore be viable alternatives to existing clustering problems which are NP‐hard, offering the advantage of efficient algorithms.
We study the computational complexity and approximability for the problem of partitioning a vertex-weighted undirected graph into p connected subgraphs with minimum gap between the largest and the smallest vertex weights.
Precise land leveling of farmlands is an important foundation of modern ground irrigation system. It can improve the uniformity of irrigation as well as soil water and salt distribution, thus controlling weeds, saving water and increasing yield. The application of GNSS (global navigation satellite system)-controlled land leveling technology is protected from weather. Besides, it has high speed and working efficiency, which earns broad application prospects. Combined with the actual situation of land leveling in China as well as the early research work, based on GNSS-controlled land leveling technology, the paper developed a leveling system which is suitable for China. The system reduces the cost of equipment and optimizes its leveling function. To reduce the number of discrete devices and improve system integration, the paper designed an integrated control terminal, including an integrated GNSS receiver module, a differential radio module, a core processing platform and a valve control module. The paper also designed the outer covering of integrated control terminal. It can be rotated to adjust the bracket through the tractor cab, and it is protected from dust, making it convenient to operate. To optimize topographic survey, the paper acquired more terrain information, for instance, the area of farmlands, which guides the later leveling operations. It also adds function of real-time display of measurement tracks. What's more, it optimizes the reference design to achieve any grade slope leveling by building a mathematic model and solve it. The optimized system consists of integrated intelligent control terminals, independent differential GNSS receiver module, hydraulic system and flat shovel equipment. Meanwhile, based on Visual Studio 2008 as the development environment, the integrated software helps a lot to land leveling. The software has accomplished the function of topographic survey, reference design and leveling operation. The intelligent precise leveling system has been applied for long experiment on Shang Zhuang experimental station of China Agricultural University. The accuracy of the system is encouraging: the maximum elevation difference after the plane leveling goes down from 20.9 cm to 10 cm and height standard deviation of elevation drops from 10.6 cm to 5.5 cm; the cumulative percentage, of which the accuracy errors are less than 5 cm, is increased from 77% to 90% after the land leveling. Besides, flat slope gradient goes down from 0.239% to 0.120%, which is in line with the design requirements. The results showed that the system is stable and reliable for operation and it perfectly meets the requirements of precise land leveling. The system developed in this paper is suitable for our farmland and farm equipment with abundant function and is easy to operate while the cost is reduced by more than a half compared with the similar foreign advanced products. It will help as fine irrigation and water-saving equipment in China and it will earn large-scale application in our country. ©, 2015, Chinese Society of Agricultural Engineering. All right reserved.
We consider a connected graph G with n vertices, p of which are centers, while the remaining ones are units. For each unit-center pair, there is a fixed assignment cost and for each vertex there is a nonnegative weight. In this article, we study the problem of partitioning G into p connected components such that each component contains exactly one center (p-centered partition). We analyze different optimization problems of this type by defining different objective functions based on the assignment costs, or on the vertices' weights, or on both of them. For these problems, we show that they are NP-hard on very special classes of graphs, and for some of them we provide polynomial time algorithms when G is a tree. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015
The sectorization problem is a particular case of partitioning problems occurring in cartography. The aim is to partition a territory into sectors such that the statistical activity measure of each sector is as close as possible to a given target value. We model this as a problem of minimizing the maximum deviation among all the sectors between their activity measure and their target value. We propose a mathematical programming formulation for the problem, we add some valid inequalities to restrict the solution space and develop a preprocessing procedure to reduce the number of variables. Computational results on different maps highlight the strong efficiency of this reduction procedure.
Methods for constructing simultaneous confidence intervals for all possible linear contrasts among several means of normally distributed variables have been given by Scheffé and Tukey. In this paper the possibility is considered of picking in advance a number (say m) of linear contrasts among k means, and then estimating these m linear contrasts by confidence intervals based on a Student t statistic, in such a way that the overall confidence level for the m intervals is greater than or equal to a preassigned value. It is found that for some values of k, and for m not too large, intervals obtained in this way are shorter than those using the F distribution or the Studentized range. When this is so, the experimenter may be willing to select the linear combinations in advance which he wishes to estimate in order to have m shorter intervals instead of an infinite number of longer intervals.
The comparison of two treatments generally falls into one of the following two categories: (a) we may have a number of replications for each of the two treatments, which are unpaired, or (b) we may have a number of paired comparisons leading to a series of differences, some of which may be positive and some negative. The appropriate methods for testing the significance of the differences of the means in these two cases are described in most of the textbooks on statistical methods.
Experimental analysis of the performance of a proposed method is a crucial and necessary task in an investigation. In this paper, we focus on the use of nonparametric statistical inference for analyzing the results obtained in an experiment design in the field of computational intelligence. We present a case study which involves a set of techniques in classification tasks and we study a set of nonparametric procedures useful to analyze the behavior of a method with respect to a set of algorithms, such as the framework in which a new proposal is developed.Particularly, we discuss some basic and advanced nonparametric approaches which improve the results offered by the Friedman test in some circumstances. A set of post hoc procedures for multiple comparisons is presented together with the computation of adjusted p-values. We also perform an experimental analysis for comparing their power, with the objective of detecting the advantages and disadvantages of the statistical tests described. We found that some aspects such as the number of algorithms, number of data sets and differences in performance offered by the control method are very influential in the statistical tests studied. Our final goal is to offer a complete guideline for the use of nonparametric statistical procedures for performing multiple comparisons in experimental studies.
We consider the k-balanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting difierent compo- nents. We allow k to be part of the input and denote the cardinality of the vertex set by n. This problem is a natural and important generalization of well-known graph partitioning problems, including minimum bisec- tion and minimum balanced cut. We present a (bi-criteria) approximation algorithm achieving an approximation of O( p lognlogk), which matches or improves over previous algorithms for all relevant values of k. Our algorithm uses a semidefl- nite relaxation which combines '22 metrics with spread- ing metrics. Surprisingly, we show that the integrality gap of the semideflnite relaxation is ›(logk) even for large values of k (e.g., k = n›(1)), implying that the de- pendence on k of the approximation factor is necessary. This is in contrast to previous approximation algorithms for k-balanced partitioning, which are based on linear programming relaxations and their approximation fac- tor is independent of k.
Let Q be a vertex-weighted path with n vertices. For any pair (L,U) can one find a partition of Q into (a given number p of) subpaths, such that the total weight of every subpath lies between L and U? We present linear-time algorithms for the partitioning problem for given (L,U) and an ) algorithm, relying on the above procedures, for finding a partition that minimizes the difference between the largest and the smallest weight of a subpath (most uniform partitioning). Our approach combines a preprocessing procedure, which detects “obstructions”, if any, via a sequence of vertex compressions; and a greedy procedure, which actually finds the desired partition. Path partitioning can be a useful tool in facing image degradation. In fact whenever a picture is taken or converted from one form to another, the resulting image can be affected by different types and degrees of degradation; if we have no informations on the actual degradation process that has taken place on the image (or if it is too difficult or costly to find such informations), the only way for image enhancement consists in increasing contrast and reducing noise by suitable modifications of the grey level of pixels. Finding the optimal grey scale transformation which leads to this enhancement can be formulated as the problem of partitioning into connected components a path with vertices corresponding to grey levels and vertex weights equal to the number of occurrences of the corresponding tone in the image, so that the sum of the weights of the vertices in each component is “as constant as possible”. In addition to image processing, this problem has applications in paging, clustering and the design of communication networks.
This paper presents a hybrid of a general heuristic framework and a general purpose mixed-integer programming (MIP) solver. The framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched. A mixed integer programming solver and its built-in feasibility heuristics is used to search a neighborhood for improving solutions. The general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods. The hybrid heuristic framework is applied to the multi-item capacitated lot sizing problem with setup times, where experiments have been conducted on a series of instances from the literature and a newly generated extension of these. On average the presented heuristic outperforms the best heuristics from the literature, and the upper bounds found by the commercial MIP solver ILOG CPLEX using state-of-the-art MIP formulations. Furthermore, we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature.
Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are given integers such that 0≤l≤u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such a partition is called an (l,u)-partition. We deal with three problems to find an (l,u)-partition of a given graph: the minimum partition problem is to find an (l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l,u)-partition with a given number p of components. All these problems are NP-hard even for series-parallel graphs, but are solvable in linear time for paths.
In this paper, we present the first polynomial-time algorithm to solve the three problems for arbitrary trees.
The basic problem considered is that of interconnecting a given set of terminals with a shortest possible network of direct links. Simple and practical procedures are given for solving this problem both graphically and computationally. It develops that these procedures also provide solutions for a much broader class of problems, containing other examples of practical interest.
This paper attempts to review and expand upon the relationship between graph theory and the clustering of a set of objects. Several graphtheoretic criteria are proposed for use within a general clustering paradigm as a means of developing procedures in between the extremes of complete-link and single-link hierarchical partitioning; these same ideas are then extended to include the more general problem of constructing subsets of objects with overlap. Finally, a number of related topics are surveyed within the general context of reinterpreting and justifying methods of clustering either through standard concepts in graph theory or their simple extensions.
The k -cut problem is to find a partition of an edge
weighted graph into k nonempty components, such that the total
edge weight between components is minimum. This problem is NP-complete
for arbitrary k and its version involving fixing a vertex in
each component is NP hard even for k =3. A polynomial algorithm
for the case of a fixed k is presented
Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs. 1.
We propose performance profiles-distribution functions for a performance metric-as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation.
Approximability of the maximally balanced connected partition problem in graphs
- Chlebikova
Partitioning a weighted graph to connected subgraphs of almost uniform size
- Ito
Some polynomial special cases for the minimum gap graph partitioning problem
- Bruglieri