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15
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 958 (14 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
Solving Quadratic (0,1)Problems by Semidefinite Programs and Cutting Planes
, 1996
"... We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moder ..."
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Cited by 51 (7 self)
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We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner.
Nonpolyhedral Relaxations of GraphBisection Problems
, 1993
"... We study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing node and edgeincidence vectors of cuts. We prove that both relaxations provide the same bound. The main fact we prove is that the duality be ..."
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Cited by 39 (8 self)
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We study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes. We formulate two lower bounds on the problem by relaxing node and edgeincidence vectors of cuts. We prove that both relaxations provide the same bound. The main fact we prove is that the duality between the relaxed edge and nodevectors preserves very natural cardinality constraints on cuts. We present an analogous result also for the maxcut problem, and show a relation between the edge relaxation and some other optimality criteria studied before. Finally, we briefly mention possible applications for a practical computational approach.
A Randomized Approximation Scheme for Metric MAXCUT
"... Metric MAXCUT is the problem of dividing a set of points in metric space into two parts so as to maximize the sum of the distances between points belonging to distinct parts. We show that metric MAXCUT has a polynomial time randomized approximation scheme. 1. Introduction 1.1. Background MAXCUT, ..."
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Cited by 30 (4 self)
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Metric MAXCUT is the problem of dividing a set of points in metric space into two parts so as to maximize the sum of the distances between points belonging to distinct parts. We show that metric MAXCUT has a polynomial time randomized approximation scheme. 1. Introduction 1.1. Background MAXCUT, the problem of finding a 2partition of the vertices of a (possibly weighted) graph which maximizes the number of edges (or sum of edge weights) accross the partition, has recently attracted a lot of attention. It has been known for a long time that this basic optimization problem is NPhard [8] but has a (straightforward) .5approximation algorithm [18]. The best approximation in the general case is a recent exciting .87856approximation algorithm due to Goemans and Williamson [9, 10], building upon previous work [4, 5, 15]. Unfortunately, there is not much room for improvement since the problem is Max SNPhard [14], and hence [17] has no fflapproximation scheme if P 6= NP. Thus one is ...
Solving MaxCut to Optimality by Intersecting Semidefinite and Polyhedral Relaxations
, 2007
"... We present a method for finding exact solutions of MaxCut, the problem of finding a cut of maximum weight in a weighted graph. We use a BranchandBound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly opti ..."
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Cited by 21 (3 self)
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We present a method for finding exact solutions of MaxCut, the problem of finding a cut of maximum weight in a weighted graph. We use a BranchandBound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal ” solution of the basic semidefinite MaxCut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the MaxCut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 01 optimization and to instances of the graph equipartition problem. The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. 1 The MaxCut Problem The MaxCut problem is one of the basic NPhard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g.,
Design and Performance of Parallel and Distributed Approximation Algorithms for Maxcut
, 1995
"... We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in a weighted undirected graph. Our implementation starts with the recent (serial) algorithm of Goemans and Williamson for this problem. We consider several different versions of this algorithm, varying the ..."
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Cited by 17 (0 self)
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We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in a weighted undirected graph. Our implementation starts with the recent (serial) algorithm of Goemans and Williamson for this problem. We consider several different versions of this algorithm, varying the interiorpoint part of the algorithm in order to optimize the parallel efficiency of our method. Our work aims for an efficient, practical formulation of the algorithm with closeto optimal parallelization. We analyze our parallel algorithm in the LogP model and predict linear speedup for a wide range of the parameters. We have implemented the algorithm using the message passing interface (MPI) and run it on several parallel machines. In particular, we present performance measurements on the IBM SP2, the Connection Machine CM5, and a cluster of workstations. We observe that the measured speedups are predicted well by our analysis in the LogP model. Finally, we test our implementation on s...
Semidefinite Programming and Combinatorial Optimization
 Appl. Numer. Math
, 1998
"... Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete opti ..."
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Cited by 12 (4 self)
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Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete optimization problems. The duality theory for semidefinite programs is the key to understand algorithms to solve them. The relevant duality results are therefore summarized. The second part of the paper deals with the approximation of integer problems both in a theoretical setting, and from a computational point of view. 1 Introduction The interest in Semidefinite Programming (SDP) has been growing rapidly in the last few years. Here are some possible explanations for this sudden rise of interest. The algorithmic development of interiorpoint methods for Linear Programs indicated the potential of this approach to solve general convex problems. Semidefinite Programs are a natural generaliza...
Path Optimization for Graph Partitioning Problems
 Discrete Applied Mathematics. Combinatorial Algorithms, Optimization and Computer Science
, 1998
"... This paper presents a new heuristic for graph partitioning called Path Optimization (PO), and the results of an extensive set of empirical comparisons of the new algorithm with two very wellknown algorithms for partitioning: the KernighanLin algorithm and simulated annealing. Our experiments are ..."
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Cited by 11 (2 self)
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This paper presents a new heuristic for graph partitioning called Path Optimization (PO), and the results of an extensive set of empirical comparisons of the new algorithm with two very wellknown algorithms for partitioning: the KernighanLin algorithm and simulated annealing. Our experiments are described in detail, and the results are presented in such a way as to reveal performance trends based on several variables. Sufficient trials are run to obtain 99% confidence intervals small enough to lead to a statistical ranking of the implementations for various circumstances. The results for geometric graphs, which have become a frequentlyused benchmark in the evaluation of partitioning algorithms, show that PO holds an advantage over the others. In addition to the main test suite described above, comparisons of PO to more recent partitioning approaches are also given. We present the results of comparisons of PO with a parallelized implementation of Goemans' and Williamson's 0.878 appr...
Combinatorial approximation algorithms  Guaranteed versus experimental performance
, 2002
"... ..."
Node and Edge Relaxations of the MaxCut Problem
, 1994
"... We study an upper bound on the maxcut problem defined via a relaxation of the discrete problem to a continuous nonlinear convex problem, which can be solved efficiently. We demonstrate how far the approach can be pushed using advanced techniques from numerical linear algebra and nonsmooth optimizat ..."
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Cited by 6 (2 self)
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We study an upper bound on the maxcut problem defined via a relaxation of the discrete problem to a continuous nonlinear convex problem, which can be solved efficiently. We demonstrate how far the approach can be pushed using advanced techniques from numerical linear algebra and nonsmooth optimization. Various classes of graphs with up to 50,000 nodes and up to four million edges are considered. Since the theoretical bound can be computed only with a certain precision in practice, we use duality between node and edgeoriented relaxations to estimate the difference between the theoretical and the computed bounds.