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17
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
On the Notion of Interestingness in Automated Mathematical Discovery
 International Journal of Human Computer Studies
, 2000
"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathema ..."
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Cited by 66 (25 self)
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We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.
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.
ILP for Mathematical Discovery
 In Proceedings of the 13th International Conference on Inductive Logic Programming
, 2003
"... We believe that AI programs written for discovery tasks will need to simultaneously employ a variety of reasoning techniques such as induction, abduction, deduction, calculation and invention. We describe the HR system which performs a novel ILP routine called automated theory formation. This co ..."
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Cited by 9 (3 self)
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We believe that AI programs written for discovery tasks will need to simultaneously employ a variety of reasoning techniques such as induction, abduction, deduction, calculation and invention. We describe the HR system which performs a novel ILP routine called automated theory formation. This combines inductive and deductive reasoning to form clausal theories consisting of classi cation rules and association rules.
Making Conjectures about Maple Functions
 In: Proceedings of the Tenth Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, LNAI 2385
, 2002
"... One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand. ..."
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Cited by 9 (6 self)
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One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand.
An Applicationbased Comparison of Automated Theory Formation and Inductive Logic Programming
 Linkoping Electronic Articles in Computer and Information Science (special issue: Proceedings of Machine Intelligence
, 2000
"... Automated theory formation involves the production of examples, concepts and hypotheses about the concepts. The HR program performs automated theory formation and has used to form theories in mathematical domains. In addition to providing a plausible model for automated theory formation, HR has been ..."
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Cited by 5 (5 self)
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Automated theory formation involves the production of examples, concepts and hypotheses about the concepts. The HR program performs automated theory formation and has used to form theories in mathematical domains. In addition to providing a plausible model for automated theory formation, HR has been applied to some applications in machine learning. We discuss HR's application to inducing de nitions from examples, scienti c discovery, problem solving and puzzle generation. For each problem, we look at how theory formation was applied, and mention some initial results from using HR.
Intelligent Machinery and Mathematical Discovery
 Science
, 1999
"... All published research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described for the first time. The fundamental principle underlying ..."
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Cited by 4 (0 self)
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All published research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described for the first time. The fundamental principle underlying this program can be simply stated: make the strongest conjecture for which no counterexample is known. Conjecturemaking may be key to building machines with a wide variety of intelligent behaviors. If so, this principle should prove exceptionally useful.
The NumbersWithNames Program
 PROCEEDINGS OF THE SEVENTH AI AND MATHS SYMPOSIUM
, 2002
"... We present the NumbersWithNames program which performs datamining on the Encyclopedia of Integer Sequences to nd interesting conjectures in number theory. The program forms conjectures by nding empirical relationships between a sequence chosen by the user and those in the Encyclopedia. Furthermore, ..."
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Cited by 3 (3 self)
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We present the NumbersWithNames program which performs datamining on the Encyclopedia of Integer Sequences to nd interesting conjectures in number theory. The program forms conjectures by nding empirical relationships between a sequence chosen by the user and those in the Encyclopedia. Furthermore, it transforms the chosen sequence into another set of sequences about which conjectures can also be formed. Finally, the program prunes and sorts the conjectures so that the most plausible ones are presented rst. We describe here the many improvements to the previous Prolog implementation which have enabled us to provide NumbersWithNames as an online program. We also present some new results from using NumbersWithNames, including details of an
An Updated Survey of Research in Automated Mathematical ConjectureMaking
"... This is an updated version of [33]. Research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described. The fundamental principle underlyin ..."
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Cited by 1 (0 self)
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This is an updated version of [33]. Research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described. The fundamental principle underlying this program can be simply stated: make the strongest conjecture for which no counterexample is known. Conjecturemaking may be key to building machines with a wide variety of intelligent behaviors. If so, this principle should prove exceptionally useful.