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Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Semidefinite Programming and Graph Equipartition
 In Topics in Semidefinite and InteriorPoint Methods
, 1998
"... . Semidefinite relaxations are used to approximate the problem of partitioning a graph into equally sized components. The relaxations extend previous eigenvalue based models, and combine semidefinite and polyhedral approaches. Computational results on graphs with several hundred vertices are given, ..."
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Cited by 23 (7 self)
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. Semidefinite relaxations are used to approximate the problem of partitioning a graph into equally sized components. The relaxations extend previous eigenvalue based models, and combine semidefinite and polyhedral approaches. Computational results on graphs with several hundred vertices are given, and indicate that semidefinite relaxations approximate the equipartition problem quite well. 1 Introduction Semidefinite Programming has turned out to be a powerful tool in the analysis of heuristics for difficult graph optimization problems. Lov'asz [1979] has used it to approximate the clique number and the chromatic number of graphs, introducing the theta function `(G) of a graph G. More recently Goemans and Williamson [1995] showed that a certain heuristic of the maxcut problem, which is based on a semidefinite program, produces a cut with weight guaranteed to be within about 14% of the optimum value. This result initiated a sequence of papers using semidefinite programs to analyze ap...
An analysis of convex relaxations for MAP estimation of discrete MRFs
 Journal of Machine Learning Research
, 2009
"... The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NPhard. However, due to its central importance in many applications, several approximation algorithms have been pr ..."
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Cited by 14 (0 self)
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The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NPhard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LPS: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in Chekuri et al. (2001), Koster et al. (1998), and Wainwright et al. (2005) for the general case; (ii) QPRL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCPMS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by Kumar et al. (2006) for a general label set. We show that the SOCPMS and the QPRL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints/objective function offered by QP and SOCP, the LPS relaxation strictly dominates (i.e., provides a better approximation than) QPRL and SOCPMS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations
{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...
Orbitopal fixing
 Integer Programming and Combinatorial Optimization, Proceedings of the Twelfth International IPCO Conference, volume 4513 of LNCS
, 2007
"... Abstract. The topic of this paper are integer programming models in which a subset of 0/1variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branchandcut algorithms if the order of the subsets of the partition is irrelevant. ..."
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Abstract. The topic of this paper are integer programming models in which a subset of 0/1variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branchandcut algorithms if the order of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branchandcut tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branchandcut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the branchandcut tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been investigated in [11]. It does, however, not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks. 1.