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The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 140 (20 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Solving Hard Set Covering Problems
 Operations Research Letters
, 1995
"... We propose a new branchandbound algorithm to solve hard instances of set covering problems arising from Steiner triple systems. 1 Introduction. The set covering probem (SC) is the problem of finding the minimum number of elements of a ground set E intersecting each member of a given family of s ..."
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Cited by 21 (1 self)
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We propose a new branchandbound algorithm to solve hard instances of set covering problems arising from Steiner triple systems. 1 Introduction. The set covering probem (SC) is the problem of finding the minimum number of elements of a ground set E intersecting each member of a given family of subsets of E. (SC) is known to be NPhard ([14]). Several algorithms have been studied and implemented in order to solve (SC), exactly or approximately, see [3, 4, 7, 8, 10, 11, 12, 13, 16]. In 1974 Fulkerson, Nemhauser and Trotter [13] described a class of computationally difficult set covering problems arising from a class of set systems known as Steiner triple systems (STS), and they suggested that these problems could be used as benchmarks to test the quality of different algorithms for the set covering problem. In particular, they introduced four special instances of (STS) with 9, 15, 27 and 45 elements; we denote them by STS 9 , STS 15 , STS 27 and STS 45 . They were able to solve STS 9...
An Algorithm for Finding a Maximum Clique in a Graph
, 1997
"... This paper introduces a branchandbound algorithm for the maximum clique problem which applies existing clique finding and vertex coloring heuristics to determine lower and upper bounds for the size of a maximum clique. Computational results on a variety of graphs indicate the proposed procedure in ..."
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Cited by 21 (0 self)
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This paper introduces a branchandbound algorithm for the maximum clique problem which applies existing clique finding and vertex coloring heuristics to determine lower and upper bounds for the size of a maximum clique. Computational results on a variety of graphs indicate the proposed procedure in most instances outperforms leading algorithms.
Structural Similarity and Adaptation
, 1996
"... Most commonly, casebased reasoning is applied in domains where attribute value representations of cases are sufficient to represent the features relevant to support classification, diagnosis or design tasks. Distance functions like the Hammingdistance or their transformation into similarity functi ..."
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Cited by 11 (0 self)
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Most commonly, casebased reasoning is applied in domains where attribute value representations of cases are sufficient to represent the features relevant to support classification, diagnosis or design tasks. Distance functions like the Hammingdistance or their transformation into similarity functions are applied to retrieve past cases to be used to generate the solution of an actual problem. Often, domain knowledge is available to adapt past solutions to new problems or to evaluate solutions. However, there are domains like architectural design or law in which structural case representations and corresponding structural similarity functions are needed. Often, the acquisition of adaptation knowledge seems to be impossible or rather requires an effort that is not manageable for fielded applications. Despite of this, humans use cases as the main source to generate adapted solutions. How to achieve this computationally? This paper presents a general approach to structural similarity as...
An Exact Algorithm For The Maximum Stable Set Problem
 Computational Optimization and Application
, 1994
"... We describe a new branchandbound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different nodefixing heuristics are also described. Computational te ..."
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Cited by 11 (2 self)
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We describe a new branchandbound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different nodefixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to `reallife' problems show that the algorithm is competitive with the fastest algorithms known so far. 1 Introduction We denote by G = (V; E) an undirected graph. V is the set of nodes and E the set of edges. A stable set is a subset of V such that no two nodes of the subset are pairwise adjacent. The cardinality of a maximum stable set of G will be denoted by ff(G). A clique is a subset of V with the property that all the nodes are pairwise adjacent. A clique covering is a set of disjoint cliques whose union is equal to V ; the cardinality of a minimum clique covering is denoted by `(G), and since at most one nod...
An Optimisation Algorithm for Maximum Independent Set with Applications in Map Labelling
, 1999
"... . We consider the following map labelling problem: given distinct points p1 ; p2 ; : : : ; pn in the plane, nd a set of pairwise disjoint axisparallel squares Q1 ; Q2 ; : : : ; Qn where p i is a corner of Q i . This problem reduces to that of nding a maximum independent set in a graph. We presen ..."
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Cited by 10 (1 self)
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. We consider the following map labelling problem: given distinct points p1 ; p2 ; : : : ; pn in the plane, nd a set of pairwise disjoint axisparallel squares Q1 ; Q2 ; : : : ; Qn where p i is a corner of Q i . This problem reduces to that of nding a maximum independent set in a graph. We present a branch and cut algorithm for nding maximum independent sets and apply it to independent set instances arising from map labelling. The algorithm uses a new technique for setting variables in the branch and bound tree that implicitly exploits the Euclidean nature of the independent set problems arising from map labelling. Computational experiments show that this technique contributes to controlling the size of the branch and bound tree. We also present a novel variant of the algorithm for generating violated oddhole inequalities. Using our algorithm we can nd provably optimal solutions for map labelling instances with up to 950 cities within modest computing time, a considera...
Edge Projection and the Maximum Cardinality Stable Set Problem
, 1996
"... . Edge projection is a specialization of Lov'asz and Plummer's clique projection when restricted to edges. We discuss some properties of the edge projection which are then exploited to develop a new upper bound procedure for the Maximum Cardinality Stable Set Problem (MSS Problem). The upper bound c ..."
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Cited by 9 (1 self)
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. Edge projection is a specialization of Lov'asz and Plummer's clique projection when restricted to edges. We discuss some properties of the edge projection which are then exploited to develop a new upper bound procedure for the Maximum Cardinality Stable Set Problem (MSS Problem). The upper bound computed by our heuristic, incorporated in a branchandbound scheme in conjunction with Balas and Yu branching rule, seems to be very effective for sparse graphs, which are typically hard instances of the MSS Problem. 1. Introduction We denote by G = (V; E) an undirected graph. V is the set of nodes and E the set of edges. A stable set is a subset of V such that no two nodes of the subset are pairwise adjacent. The cardinality of a maximum stable set of G will be denoted by ff(G). When G is the empty graph, ff(G) = 0. A clique is a subset of V with the property that all the nodes are pairwise adjacent. A clique partitioning of G is a family of cliques such that each node of G is contained ...
APPROXIMATING MAXIMUM STABLE SET AND MINIMUM GRAPH COLORING PROBLEMS WITH THE POSITIVE SEMIDEFINITE RELAXATION
"... We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in m ..."
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Cited by 9 (1 self)
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We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.
Automatic Indexing, Retrieval and Reuse of Topologies in Architectual Layouts
 StudioProceedings of the 6th International Conference on ComputerAided Architectural Design Futures
, 1995
"... Former layouts contain much of the knowhow of architects. A generic and automatic way to formalize this knowhow in order to use it by a computer would save a lot of effort and money. However, there seems to be no such way. The only access to the knowhow are the layouts themselves. Developing a ge ..."
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Cited by 7 (2 self)
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Former layouts contain much of the knowhow of architects. A generic and automatic way to formalize this knowhow in order to use it by a computer would save a lot of effort and money. However, there seems to be no such way. The only access to the knowhow are the layouts themselves. Developing a generic software tool to reuse former layouts you cannot consider every part of the architectual domain or things like personal style. Tools used today only consider small parts of the architectual domain. Any personal style is ignored. Isn't it possible to build a basic tool which is adjusted by the content of the former layouts, but may be extended incremently by modeling as much of the domain as desirable? This paper will describe a reuse tool to perform this task focusing on topological and geometrical binary relations. 1 Introduction The main idea of TOPO is to transfer the knowledge about useful topologies from former layouts to a new problem in order to correct, extend or detail the qu...