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Variable Neighborhood Search
, 1997
"... Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. In this survey paper we present basic rules of VNS and some of its extensions. Moreover, applications a ..."
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Cited by 242 (24 self)
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Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. In this survey paper we present basic rules of VNS and some of its extensions. Moreover, applications are briefly summarized. They comprise heuristic solution of a variety of optimization problems, ways to accelerate exact algorithms and to analyze heuristic solution processes, as well as computerassisted discovery of conjectures in graph theory.
Variable neighborhood search: Principles and applications
, 2001
"... Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using an ..."
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Cited by 119 (15 self)
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Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a twolevel VNS, called variable neighborhood decomposition search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear
Variable Neighborhood Search for Extremal Graphs 6. Analyzing Bounds for the Connectivity Index
, 2000
"... Recently, Araujo and De la Pe~na [1] gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which ..."
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Cited by 23 (7 self)
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Recently, Araujo and De la Pe~na [1] gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Pe\~na [1] are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of given order and number of pending vertices with minimum and with maximum connectivity index are characterized.
A Tutorial on Variable Neighborhood Search
 LES CAHIERS DU GERAD, HEC MONTREAL AND GERAD
, 2003
"... Variable Neighborhood Search (VNS) is a recent metaheuristic, or framework for building heuristics, which exploits systematically the idea of neighborhood change, both in the descent to local minima and in the escape from the valleys which contain them. In this tutorial we first present the ingre ..."
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Cited by 16 (3 self)
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Variable Neighborhood Search (VNS) is a recent metaheuristic, or framework for building heuristics, which exploits systematically the idea of neighborhood change, both in the descent to local minima and in the escape from the valleys which contain them. In this tutorial we first present the ingredients of VNS, i.e., Variable Neighborhood Descent (VND) and Reduced VNS (RVNS) followed by the basic and then the general scheme of VNS itself which contain both of them. Extensions are presented, in particular Skewed VNS (SVNS) which enhances exploration of far away valleys and Variable Neighborhood Decomposition Search (VNDS), a twolevel scheme for solution of large instances of various problems. In each case, we present the scheme, some illustrative examples and questions to be addressed in order to obtain an efficient implementation.
Eigenvalue bounds for the signless Laplacian
 Publ. Inst. Math. (Beograd
"... Abstract. We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are present ..."
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Cited by 11 (0 self)
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Abstract. We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computergenerated conjectures. 1.
A Variable Neighbourhood Monte Carlo Search For Component Placement Sequencing Of Multi Headed Placement Machine
 SUBMITTED TO JOURNAL OF INTELLIGENT MANUFACTURING
, 2005
"... ..."
inequalities among graph invariants: using GraPHedron to uncover optimal relationships. Accepted for publication in Networks (2008
"... Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph nodes, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tupl ..."
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Cited by 4 (3 self)
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Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph nodes, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a “small ” number of nodes. It greatly helps conjecturing optimal linear inequalities, which are then hopefully proved for any node number. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities in the stability number and the link number are obtained. To this aim, a problem of Ore (1962) related to Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter.
G.: An iterative restart variable neighbourhood search for the examination timetabling problem
 Practice and Theory of Automated Timetabling. LNCS
, 1996
"... Producing good quality examination timetables is a difficult task which is faced by many academic institutions. Due to the complexity and the large size of the realworld university examination timetabling problems, it is difficult to obtain an optimal solution. Indeed, due to the complex nature of ..."
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Cited by 4 (4 self)
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Producing good quality examination timetables is a difficult task which is faced by many academic institutions. Due to the complexity and the large size of the realworld university examination timetabling problems, it is difficult to obtain an optimal solution. Indeed, due to the complex nature of the problem, it is questionable if an end user would recognise a truly optimal solution.
A Survey of Research on Automated Mathematical ConjectureMaking
 FAJTLOWICZ (EDITORS), AMERICAN MATHEMATICAL SOCIETY
, 2005
"... The first attempt at automating mathematical conjecturemaking appeared in the late1950s. It was not until the mid1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlyi ..."
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Cited by 3 (0 self)
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The first attempt at automating mathematical conjecturemaking appeared in the late1950s. It was not until the mid1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest Conjecture: make the strongest conjecture for which no counterexample is known. These two programs as well as other attempts to automate mathematical conjecturemaking are surveyed—the success of a conjecturemaking program, it is found, correlates strongly whether the program is designed to produce statements that are relevant to answering or advancing our mathematical questions.
Trees of extremal connectivity index
 Discrete Appl. Math
"... The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v)) α of all edges uv of G, where α is a real number (α � = 0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T ..."
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Cited by 3 (1 self)
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The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v)) α of all edges uv of G, where α is a real number (α � = 0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T). Also, for −1 ≤ α < 0, we give a sharp lower bound and a upper bound for wα(T).