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56
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 936 (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 ...
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 202 (18 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 42 (12 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 41 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.
Statistical Mechanics, ThreeDimensionality and NPcompleteness I. Universality of Intractability for the Partition Function of the Ising Model Across NonPlanar Lattices (Extended Abstract)
"... This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working ..."
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Cited by 28 (1 self)
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This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working conjectures in the statistical mechanics literature, such as the CrossedBonds conjecture, and the impossibility to compute effectively the partition functions for any threedimensional lattice Ising model � these conjectures apply to the Onsager algebraic method, the Fermion operators method, and the combinatorial method based on Pfaffians. The fundamental results of the area, including those of Onsager, Kac, Feynman, Fisher, Kasteleyn, Temperley, Green, Hurst and more recently Barahona: for every Planar crystal lattice the partition functions for the nite sublattices can be computed in polynomialtime, paired with the results of this paper: for every NonPlanar crystal lattice computing the parition functions for the finite sublattices is NPcomplete, provide an exact characterization for several of the most studied Ising models. Our results settle at once, for several models, (1) the 2D nonplanar vs. 2D planar, (2) the nextnearest neighbour
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 25 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
On a Positive Semidefinite Relaxation of the Cut Polytope
, 1993
"... We study the convex body e Ln defined by e Ln := fX j X = (x ij ) positive semidefinite n \Theta n matrix ; x ii = 1 for all ig: Our main motivation for investigating this body comes from combinatorial optimization, namely from approximating the maxcut problem. An important property of e Ln is th ..."
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Cited by 25 (5 self)
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We study the convex body e Ln defined by e Ln := fX j X = (x ij ) positive semidefinite n \Theta n matrix ; x ii = 1 for all ig: Our main motivation for investigating this body comes from combinatorial optimization, namely from approximating the maxcut problem. An important property of e Ln is that, due to the positive semidefinite constraints, one can optimize over it in polynomial time. On the other hand, e Ln still inherits the difficult structure of the underlying combinatorial problem. In particular, it is NPhard to decide whether the optimum of the problem minTr(CX); X 2 e Ln is reached in a vertex. This result follows from the complete characterization of the matrices C of the form C = bb t for some vector b, for which the optimum of the above program is reached in a vertex. We describe several geometric properties of e Ln . Among other facts, we show that e Ln has 2 n\Gamma1 vertices corresponding to all bipartitions of the set f1; 2; : : :; ng.
Exact ground states of Ising spin glasses: New experimental results with a branch and cut algorithm
, 1995
"... In this paper we study 2dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100 ..."
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Cited by 24 (3 self)
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In this paper we study 2dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100 x 100 can be determined in a moderate amount of computation time, and we report on extensive computational tests. With our method we produce results based on more than 20 000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other more popular ways to compute the ground state, like Monte Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determined a priori. Our ground state energy estimation at zero field is 1.317.
SEPARATING A SUPERCLASS OF COMB INEQUALITIES IN PLANAR GRAPHS
, 2000
"... Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedu ..."
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Cited by 23 (6 self)
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Many classes of valid and facetinducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗ , nds one or more inequalities in the class which are violated by x , or proves that none exist. Such algorithms are at the core of the highly successful branchandcut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We de ne a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x ∗ e ¿0 is planar. The separation algorithm runs in O(n³) time, where n is the number of vertices in the graph.
PrimalDual Approximation Algorithms for Feedback Problems in Planar Graphs
 IPCO '96
, 1996
"... Given a subset of cycles of a graph, we consider the problem of finding a minimumweight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimumweight feedback vertex set problem in both directed and undirected graphs, the subset fee ..."
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Cited by 21 (3 self)
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Given a subset of cycles of a graph, we consider the problem of finding a minimumweight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimumweight feedback vertex set problem in both directed and undirected graphs, the subset feedback vertex set problem, and the graph bipartization problem, in which one must remove a minimumweight set of vertices so that the remaining graph is bipartite. We give a 9/4approximation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties. This results in 9/4approximation algorithms for the aforementioned feedback and bipartization problems in planar graphs. Our algorithms use the primaldual method for approximation algorithms as given in Goemans and Williamson [16]. We also show that our results have an interesting bearing on a conjecture of Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.