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A Lagrangian Relaxation Approach to the EdgeWeighted Clique Problem
 European Journal of Operational Research
, 1999
"... The bclique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n = CP n n as a special case and being closely related to the quadratic knapsack polytope, it has recei ..."
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Cited by 13 (0 self)
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The bclique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n = CP n n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the maxcut problem is equivalent with optimizing a linear function over CP n n . The problem of optimizing linear functions over CP n b has so far been approached via heuristic combinatorial algorithms and cuttingplane methods. We study the structure of CP n b in further detail and present a new computational approach to the linear optimization problem based on the idea of integrating cutting planes into a Lagrangian relaxation of an integer programming problem that Balas and Christofides had suggested for the traveling salesman problem. In particular, we show that the separation problem for tree inequalities becomes p...
{0, 1/2}ChvatalGomory Cuts
, 1995
"... Given the integer polyhedron P I := convfx 2 Z n : Ax bg, where A 2 Z m\Thetan and b 2 Z m , a Chv'atalGomory (CG) cut is a valid inequality for P I of the type ..."
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Cited by 12 (3 self)
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Given the integer polyhedron P I := convfx 2 Z n : Ax bg, where A 2 Z m\Thetan and b 2 Z m , a Chv'atalGomory (CG) cut is a valid inequality for P I of the type
The Interval Order Polytope of a Digraph
 In Balas & Clausen (Eds.) (1995), Proc. of the 4th Int. IPCO Conf
, 1995
"... . We introduce the interval order polytope of a digraph D as the convex hull of interval order inducing arc subsets of D. Two general schemes for producing valid inequalities are presented. These schemes have been used implicitly for several polytopes and they are applied here to the interval order ..."
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Cited by 11 (4 self)
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. We introduce the interval order polytope of a digraph D as the convex hull of interval order inducing arc subsets of D. Two general schemes for producing valid inequalities are presented. These schemes have been used implicitly for several polytopes and they are applied here to the interval order polytope. It is shown that almost all known classes of valid inequalities of the linear ordering polytope can be explained by the two classes derived from these schemes. We provide two applications of the interval order polytope to combinatorial optimization problems for which to our knowledge no polyhedral descriptions have been given so far. One of them is related to analysing DNA subsequences. 1 Introduction Interval orders and their cocomparability graphs, the interval graphs, play an important role not only in the theory of partially ordered sets and graph theory (cf., e.g., [Fis85]) but also for combinatorial optimization problems. This is due to the fact that each element is associat...
Rapid mathematical programming
, 2004
"... This book was typeset with TEX using L ATEX and many further formatting packages. The pictures were prepared using pstricks, xfig, gnuplot and gmt. All numerals in this text are recycled. Für meine Eltern Preface Avoid reality at all costs — fortune(6) As the inclined reader will find out soon enoug ..."
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Cited by 10 (2 self)
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This book was typeset with TEX using L ATEX and many further formatting packages. The pictures were prepared using pstricks, xfig, gnuplot and gmt. All numerals in this text are recycled. Für meine Eltern Preface Avoid reality at all costs — fortune(6) As the inclined reader will find out soon enough, this thesis is not about deeply involved mathematics as a mean in itself, but about how to apply mathematics to solve realworld problems. We will show how to shape, forge, and yield our tool of choice to rapidly answer questions of concern to people outside the world of mathematics. But there is more to it. Our tool of choice is software. This is not unusual, since it has become standard practice in science to use software as part of experiments and sometimes even for proofs. But in order to call an experiment scientific it must be reproducible. Is this the case?
Cliques and Clustering: A Combinatorial Approach
 Oper. Res. Lett
, 1997
"... We use column generation and a specialized branching technique for solving constrained clustering problems. We also develop and implement an innovative combinatorial method for solving the pricing subproblems. Computational experiments comparing the resulting branchandprice method to competing met ..."
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Cited by 10 (0 self)
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We use column generation and a specialized branching technique for solving constrained clustering problems. We also develop and implement an innovative combinatorial method for solving the pricing subproblems. Computational experiments comparing the resulting branchandprice method to competing methodologies in the literature are presented and suggest that our technique yields a significant improvement on the hard instances of this problem. 1 Introduction Given a graph G(V; E) where V is the vertex set and E is the edge set, and edge weights w e ; e 2 E, we consider clustering problems that involve partitioning G into connected subgraphs or clusters such that the sum of the edge weights in every cluster is maximized (or, equivalently, the total weight on edges between clusters in minimized). Several graph partitioning type problems have been studied in the literature. Uncapacitated versions of this problem model a common problem in qualitative data analysis of partitioning a number...
Clustering of microarray data via clique partitioning
 J. COMB. OPTIM
, 2005
"... Microarrays are repositories of gene expression data that hold tremendous potential for new understanding, leading to advances in functional genomics and molecular biology. Cluster analysis (CA) is an early step in the exploration of such data that is useful for purposes of data reduction, exposing ..."
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Cited by 6 (2 self)
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Microarrays are repositories of gene expression data that hold tremendous potential for new understanding, leading to advances in functional genomics and molecular biology. Cluster analysis (CA) is an early step in the exploration of such data that is useful for purposes of data reduction, exposing hidden patterns, and the generation of hypotheses regarding the relationship between genes and phenotypes. In this paper we present a new model for the clique partitioning problem and illustrate how it can be used to perform cluster analysis in this setting.
Transitive packing
 Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science 1084
, 1996
"... This paper is intended to give a concise understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope and gives cutting plane proofs for huge classes of valid inequalities of this polytope. ..."
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Cited by 4 (1 self)
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This paper is intended to give a concise understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope and gives cutting plane proofs for huge classes of valid inequalities of this polytope. We introduce generalized cycle, generalized clique, generalized antihole, generalized antiweb, generalized web, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases but give also many new inequalities for several other special cases. For some of the classes we prove as well a lower bound for their Chv'atal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices.
On Disjunctive Cuts for Combinatorial Optimization
 J. OF COMB. OPT
, 2000
"... In the successful branchandcut approach to combinatorial optimization, linear inequalities are used as cutting planes within a branchandbound framework. Although researchers often prefer to use facetinducing inequalities as cutting planes, good computational results have recently been obtaine ..."
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Cited by 4 (1 self)
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In the successful branchandcut approach to combinatorial optimization, linear inequalities are used as cutting planes within a branchandbound framework. Although researchers often prefer to use facetinducing inequalities as cutting planes, good computational results have recently been obtained using disjunctive cuts, which are not guaranteed to be facetinducing in general. A partial explanation for the success of the disjunctive cuts is given in this paper. It is shown that, for six important combinatorial optimization problems (the clique partitioning, maxcut, acyclic subdigraph, linear ordering, asymmetric travelling salesman and set covering problems), certain facetinducing inequalities can be obtained by simple disjunctive techniques. New polynomialtime separation algorithms are obtained for these inequalities as a byproduct. The disjunctive approach is then compared and contrasted with some other `generalpurpose' frameworks for generating cutting planes and some conclusions are made with respect to the potential and limitations of the disjunctive approach.
Solution Stability in Linear Programming Relaxations: Graph Partitioning and Unsupervised Learning
"... We propose a new method to quantify the solution stability of a large class of combinatorial optimization problems arising in machine learning. As practical example we apply the method to correlation clustering, clustering aggregation, modularity clustering, and relative performance significance clu ..."
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Cited by 4 (2 self)
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We propose a new method to quantify the solution stability of a large class of combinatorial optimization problems arising in machine learning. As practical example we apply the method to correlation clustering, clustering aggregation, modularity clustering, and relative performance significance clustering. Our method is extensively motivated by the idea of linear programming relaxations. We prove that when a relaxation is used to solve the original clustering problem, then the solution stability calculated by our method is conservative, that is, it never overestimates the solution stability of the true, unrelaxed problem. We also demonstrate how our method can be used to compute the entire path of optimal solutions as the optimization problem is increasingly perturbed. Experimentally, our method is shown to perform well on a number of benchmark problems. 1.