#### DMCA

## Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming (1995)

### Cached

### Download Links

- [www.almaden.ibm.com]
- [theory.lcs.mit.edu]
- [dimacs.rutgers.edu]
- [karush.rutgers.edu]
- [www-math.mit.edu]
- [www.csd.uwo.ca]
- [www-math.mit.edu]
- [luthuli.cs.uiuc.edu]
- [math.mit.edu]
- [www.convexoptimization.com]
- [luthuli.cs.uiuc.edu]
- [www-math.mit.edu]
- [www.csd.uwo.ca]
- [math.mit.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | Journal of the ACM |

Citations: | 1195 - 13 self |

### Citations

1935 |
Reducibility among combinatorial problems
- Karp
- 1972
(Show Context)
Citation Context ... For simplicity, we usually set w ij = 0 for (i; j) = 2 E and denote the weight of a cut (S;sS) by w(S;sS) = P i2S;j = 2S w ij . The MAX CUT problem is one of the Karp's original NP-complete problems =-=[37]-=-, and has long been known to be NP-complete even if the problem is unweighted; that is, if w ij = 1 for all (i; j) 2 E [19]. The MAX CUT problem is solvable in polynomial time for some special classes... |

1697 | Graph Theory with Applications - Bondy, Murthy - 1976 |

1393 |
Geometric Algorithms and Combinatorial Optimization, 2nd Edition
- Grotschel, Lovasz, et al.
- 1993
(Show Context)
Citation Context ...itive error of ffl in polynomial time (ffl is part of the input, so the running time dependence on ffl is polynomial in log 1 ffl ). This can be done through the ellipsoid algorithm (Grotschel et al. =-=[26]-=-) and other polynomial-time algorithms for convex programming (Vaidya [67]) as well as 2 interior-point methods (Nesterov and Nemirovskii [50, 51] and Alizadeh [1]). To terminate in polynomial time, t... |

1078 | Semidefinite programming
- Vandenberghe, Boyd
- 1996
(Show Context)
Citation Context ...in the design and analysis of interior-point methods for semidefinite programming; for several references available at the time of writing of this paper, see the survey paper by Vandenberghe and Boyd =-=[68]-=-. Semidefinite programming has many interesting applications in a variety of areas including control theory, nonlinear programming, geometry and combinatorial optimization; see [51, 9, 68, 1], the ref... |

792 | Proof verification and hardness of approximation problems
- Arora, Lund, et al.
- 1992
(Show Context)
Citation Context ..., MAX CUT, MAX 2SAT, and MAX DICUT are MAX SNP-hard [55], and so it is known that there exists a constant c ! 1 such that a c-approximation algorithm for any of these problems would imply that P = NP =-=[2]-=-. Bellare, Goldreich, and Sudan [6] have shown that c is as small as 83/84 for MAX CUT and 95/96 for MAX 2SAT. Since bidirected instances of MAX DICUT are equivalent to instances of MAX CUT, the bound... |

606 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ...maximum directed cut problem (MAX DICUT), where fi = min 0`!arccos(\Gamma1=3) 2 2 \Gamma 3` 1 + 3 cos ` ? 0:79607: The best previously known algorithm for MAX DICUT has a performance guarantee of 1 4 =-=[55]-=-. Our algorithm depends on a means of randomly rounding a solution to a nonlinear relaxation of the MAX CUT problem. This relaxation can either be seen as a semidefinite program or as an eigenvalue mi... |

567 |
The theory of Matrices
- Lancaster, Tismenetsky
- 1985
(Show Context)
Citation Context ...e defined over the reals. An n \Theta n matrix A is said to be positive semidefinite if for every vector x 2 R n , x T Axs0. The following statements are equivalent for a symmetric matrix A (see e.g. =-=[39]-=-): (i) A is positive semidefinite, (ii) all eigenvalues of A are non-negative, and (iii) there exists a matrix B such that A = B T B. In (iii), B can either be a (possibly singular) n \Theta n matrix,... |

542 | Primal-dual interiorpoint methods for semidefinite programming: Convergence rates, stability and numerical results
- Alizadeh, Haeberly, et al.
- 1998
(Show Context)
Citation Context ...ogramming over cones or cone-LP since the set of positive semidefinite matrices constitutes a convex cone. To some extent, semidefinite programming is very similar to linear programming; see Alizadeh =-=[1]-=- for a comparison. It inherits the very elegant duality theory of cone-LP (see Wolkowicz [70] and the exposition by Alizadeh [1]). The simplex method can be generalized to semidefinite programs (Patak... |

482 |
The ellipsoid method and its consequences in combinatorial optimization
- Grötschel, Lovász
- 1981
(Show Context)
Citation Context ...njunction with the polynomial-time solvability of semidefinite programs, this leads to the only known polynomial-time algorithm for finding the largest stable set in a perfect graph (Grotschel et al. =-=[25]-=-). More recently, there has been increased interest in semidefinite programming from a combinatorial point-of-view [46, 47, 1, 58, 17, 45]. This started with the work of Lov'asz and Schrijver [46, 47]... |

467 |
L.J.Stockmeyer Some simplified NPcomplete graph problems Theoretical
- Garey
(Show Context)
Citation Context ... ij . The MAX CUT problem is one of the Karp's original NP-complete problems [37], and has long been known to be NP-complete even if the problem is unweighted; that is, if w ij = 1 for all (i; j) 2 E =-=[19]-=-. The MAX CUT problem is solvable in polynomial time for some special classes of graphs (e.g. if the graph is planar [52, 27]). Besides its theoretical importance, the MAX CUT problem has applications... |

462 |
On the Shannon capacity of a graph
- Lovász
- 1979
(Show Context)
Citation Context ...it leads to tighter relaxations than the classical linear programming relaxations for many graph and combinatorial problems. A beautiful application of semidefinite programming is the work of Lov'asz =-=[43]-=- on the Shannon capacity of a graph. In conjunction with the polynomial-time solvability of semidefinite programs, this leads to the only known polynomial-time algorithm for finding the largest stable... |

364 | P-complete approximation problems
- Sahni, Gonzalez
- 1976
(Show Context)
Citation Context ...will also use the term "ae-approximation algorithm" for randomized polynomial-time algorithms that deliver solutions whose expected value is at least ae times the optimal. In 1976, Sahni and=-= Gonzales [66]-=- presented a 1 2 -approximation algorithm for the MAX CUT problem. Their algorithm iterates through the vertices and decides whether or not to assign vertex i to S based on which placement maximizes t... |

350 |
TSPLIB A traveling salesman problem library
- Reinelt
- 1991
(Show Context)
Citation Context ...dl [60]. In particular, we considered several different types of random graphs, as well as complete geometric graphs defined by Traveling Salesman Problem (TSP) instances from the TSPLIB (see Reinelt =-=[63]-=-). For four different types of random graphs, we ran 50 instances on graphs of 50 vertices, 20 on graphs of size 100, and 5 on graphs of size 200. In the Type A random graph, each edge (i; j) is inclu... |

343 | Cones of matrices and set-functions and 0-1 optimization
- Lovász, Schrijver
- 1991
(Show Context)
Citation Context ...lgorithm for finding the largest stable set in a perfect graph (Grotschel et al. [25]). More recently, there has been increased interest in semidefinite programming from a combinatorial point-of-view =-=[46, 47, 1, 58, 17, 45]-=-. This started with the work of Lov'asz and Schrijver [46, 47], who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programmi... |

304 | S.: Interior-point polynomial methods in convex programming - Nesterov, Nemirovsky - 1994 |

212 | Free bits, PCPs, and nonapproximability - towards tight results
- Bellare, Goldreich, et al.
- 1998
(Show Context)
Citation Context ...are MAX SNP-hard [55], and so it is known that there exists a constant c ! 1 such that a c-approximation algorithm for any of these problems would imply that P = NP [2]. Bellare, Goldreich, and Sudan =-=[6]-=- have shown that c is as small as 83/84 for MAX CUT and 95/96 for MAX 2SAT. Since bidirected instances of MAX DICUT are equivalent to instances of MAX CUT, the bound for MAX CUT also applies to MAX DI... |

210 | Approximate graph coloring by semidefinite programming
- Karger, Motwani, et al.
- 2002
(Show Context)
Citation Context ...nique to yield a .931-approximation algorithm for MAX 2SAT and a .859-approximation algorithm for MAX DICUT. By using semidefinite programming and similar rounding ideas, 3 Karger, Motwani, and Sudan =-=[36]-=- have been able to show how to color a k-colorable graph with ~ O(n 1\Gamma 3 k+1 ) colors in polynomial time. Frieze and Jerrum [18] have used the technique to devise approximation algorithms for the... |

175 | Improved approximation algorithms for MAX k-CUT
- Frieze, Jerrum
- 1997
(Show Context)
Citation Context ...ogramming and similar rounding ideas, 3 Karger, Motwani, and Sudan [36] have been able to show how to color a k-colorable graph with ~ O(n 1\Gamma 3 k+1 ) colors in polynomial time. Frieze and Jerrum =-=[18]-=- have used the technique to devise approximation algorithms for the maximum k-way cut problem that improve on the previously best known 1 \Gamma 1=k performance guarantee. Chor and Sudan [10] apply id... |

172 |
Seminumerical Algorithms, volume 2 of The Art of Computer Programming
- Knuth
- 1997
(Show Context)
Citation Context ...or providing problem instances, Joel Spencer for motivating Theorem 2.6, Farid Alizadeh, Gabor Pataki and Rob Freund for results on semidefinite programming, and Shang-Hua Teng for bringing reference =-=[38]-=- to our attention. We received other useful comments from Farid Alizadeh, Joseph Cheriyan, Jon Kleinberg, Monique Laurent, Colin McDiarmid, Giovanni Rinaldi, David Shmoys, ' Eva Tardos, and the two an... |

142 |
Approximation Algorithms for the Set Covering and Vertex Cover Problems
- Hochbaum
- 1982
(Show Context)
Citation Context ...algorithm for MAX CUT can be obtained without explicitly solving the semidefinite program. For example, the first 2approximation algorithms for weighted vertex cover involved solving a linear program =-=[32]-=-, but later Bar-Yehuda and Even [3] devised a primal-dual algorithm in which linear programming was used only in the analysis of the algorithm. Perhaps a semidefinite analog is possible for MAX CUT. T... |

140 | Approximating the value of two provers proof systems, with applications to MAX 2SAT and MAX DICUT
- Feige, Goemans
- 1995
(Show Context)
Citation Context .... Since bidirected instances of MAX DICUT are equivalent to instances of MAX CUT, the bound for MAX CUT also applies to MAX DICUT. Since the appearance of an abstract of this paper, Feige and Goemans =-=[16]-=- have extended our technique to yield a .931-approximation algorithm for MAX 2SAT and a .859-approximation algorithm for MAX DICUT. By using semidefinite programming and similar rounding ideas, 3 Karg... |

130 |
A new algorithm for minimizing convex functions over convex sets
- Vaidya
- 1989
(Show Context)
Citation Context ...nning time dependence on ffl is polynomial in log 1 ffl ). This can be done through the ellipsoid algorithm (Grotschel et al. [26]) and other polynomial-time algorithms for convex programming (Vaidya =-=[67]-=-) as well as 2 interior-point methods (Nesterov and Nemirovskii [50, 51] and Alizadeh [1]). To terminate in polynomial time, these algorithms implicitly assume some requirement on the feasible space o... |

111 |
On the approximation of maximum satisfiability
- Yannakakis
- 1994
(Show Context)
Citation Context ...amma ffl)-approximation algorithm for the maximum 2-satisfiability problem (MAX 2SAT). The best previously known algorithm for this problem has a performance guarantee of 3 4 and is due to Yannakakis =-=[71]-=-. A somewhat simpler 3 4 -approximation algorithm was given in Goemans and Williamson [22]. The improved 2SAT algorithm leads to .7584-approximation algorithm for the overall MAX SAT problem, fraction... |

107 | Two-prover one-round proof systems: Their power and their problems
- Feige, Lovász
- 1992
(Show Context)
Citation Context ...lgorithm for finding the largest stable set in a perfect graph (Grotschel et al. [25]). More recently, there has been increased interest in semidefinite programming from a combinatorial point-of-view =-=[46, 47, 1, 58, 17, 45]-=-. This started with the work of Lov'asz and Schrijver [46, 47], who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programmi... |

86 |
Finding a maximum cut of a planar graph in polynomial time
- Hadlock
- 1975
(Show Context)
Citation Context ...ete even if the problem is unweighted; that is, if w ij = 1 for all (i; j) 2 E [19]. The MAX CUT problem is solvable in polynomial time for some special classes of graphs (e.g. if the graph is planar =-=[52, 27]-=-). Besides its theoretical importance, the MAX CUT problem has applications in circuit layout design and statistical physics (Barahona et al. [4]). For a comprehensive survey of the MAX CUT problem, t... |

80 | New 3/4-approximation algorithms for the maximum satisfiability problem
- Goemans, Williamson
- 1994
(Show Context)
Citation Context ... best previously known algorithm for this problem has a performance guarantee of 3 4 and is due to Yannakakis [71]. A somewhat simpler 3 4 -approximation algorithm was given in Goemans and Williamson =-=[22]-=-. The improved 2SAT algorithm leads to .7584-approximation algorithm for the overall MAX SAT problem, fractionally better than Yannakakis ' 3 4 -approximation algorithm for MAX SAT. Finally, a slight ... |

76 |
AnApplication of Combinatorial Optimization to Statistical Physics and Circuit Layout Design
- Barahona, Grotschel, et al.
- 1988
(Show Context)
Citation Context ...al classes of graphs (e.g. if the graph is planar [52, 27]). Besides its theoretical importance, the MAX CUT problem has applications in circuit layout design and statistical physics (Barahona et al. =-=[4]-=-). For a comprehensive survey of the MAX CUT problem, the reader is referred to Poljak and Tuza [62]. Because it is unlikely that there exist efficient algorithms for NP-hard maximization problems, a ... |

76 |
879-approximation algorithms for MAX CUT and MAX 2SAT
- Goemans, Williamson
- 1994
(Show Context)
Citation Context ...experiments with the MAX CUT algorithm which show that on a number of different types of random graphs the algorithm is usually within .96 of the optimal solution. A preliminary version of this paper =-=[21]-=- presented a method to obtain deterministic versions of our approximation algorithm with the same performance guarantees. However, the method given had a subtle error, as was pointed out to us by seve... |

67 |
Problems of distance geometry and convex properties of quadratic maps
- Barvinok
- 1995
(Show Context)
Citation Context ... rank less than p 2n, and that the optimum vectors v i of (P ) can be embedded in R m with m ! p 2n. This result also follows from a more general statement about semidefinite programs due to Barvinok =-=[5]-=- and implicit in Pataki [56]: any extreme solution of a semidefinite program with k linear equalities has rank at most l where l(l+1) 2sk. 3.2 The Semidefinite Dual As mentioned in the introduction, t... |

67 | Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices
- Overton, Womersley
- 1993
(Show Context)
Citation Context ...Deltasm ns0. The equivalence of the semidefinite program we consider and the eigenvalue bound of Delorme and Poljak was established by Poljak and Rendl [58]. Building on work by Overton and Womersley =-=[54, 53]-=-, Alizadeh [1] has shown that eigenvalue minimization problems can in general be formulated as semidefinite programs. This is potentially quite useful, since there is an abundant literature on eigenva... |

65 |
A linear time approximation algorithm for the weighted vertex cover problem
- Bar-Yehuda, Even
- 1981
(Show Context)
Citation Context ...d without explicitly solving the semidefinite program. For example, the first 2approximation algorithms for weighted vertex cover involved solving a linear program [32], but later Bar-Yehuda and Even =-=[3]-=- devised a primal-dual algorithm in which linear programming was used only in the analysis of the algorithm. Perhaps a semidefinite analog is possible for MAX CUT. The second question is whether addin... |

60 |
Laplacian eigenvalues and the maximum cut problem
- Delorme, Poljak
- 1993
(Show Context)
Citation Context ... by these papers, and by the paper of Alizadeh [1]. For MAX CUT, the semidefinite programming relaxation we consider is equivalent to an eigenvalue minimization problem proposed by Delorme and Poljak =-=[13, 12]-=-. An eigenvalue minimization problem consists of minimizing a decreasing sum of the eigenvaluessi of a matrix subject to equality constraints on the matrix; that is, minimizing P i m isi , wheres1s2s\... |

48 | A geometric approach to betweenness
- Chor, Sudan
- 1998
(Show Context)
Citation Context ...d Jerrum [18] have used the technique to devise approximation algorithms for the maximum k-way cut problem that improve on the previously best known 1 \Gamma 1=k performance guarantee. Chor and Sudan =-=[10] apply ide-=-as from this paper to the "betweeness" problem. Thus it seems likely that the techniques in this paper will continue to prove useful in designing approximation algorithms. We expect that in ... |

44 | Derandomizing semidefinite programming based approximation algorithms - Mahajan, Ramesh - 1995 |

44 | Eigenvalues in combinatorial optimization - MOHAR, POLJAK - 1991 |

39 |
Self-concordant functions and polynomial time methods in convex programming
- Nesterov, Nemirovskii
- 1989
(Show Context)
Citation Context ...be done through the ellipsoid algorithm (Grotschel et al. [26]) and other polynomial-time algorithms for convex programming (Vaidya [67]) as well as 2 interior-point methods (Nesterov and Nemirovskii =-=[50, 51]-=- and Alizadeh [1]). To terminate in polynomial time, these algorithms implicitly assume some requirement on the feasible space or on the size of the optimum solution; for details see Grotschel et al. ... |

38 | Nonpolyhedral relaxations of graph-bisection problems
- POLJAK, RENDL
- 1995
(Show Context)
Citation Context ...lgorithm for finding the largest stable set in a perfect graph (Grotschel et al. [25]). More recently, there has been increased interest in semidefinite programming from a combinatorial point-of-view =-=[46, 47, 1, 58, 17, 45]-=-. This started with the work of Lov'asz and Schrijver [46, 47], who developed a machinery to define tighter and tighter relaxations of any integer program based on quadratic and semidefinite programmi... |

36 |
On the sum of the largest eigenvalues of a symmetric matrix
- Overton, Womersley
- 1992
(Show Context)
Citation Context ...Deltasm ns0. The equivalence of the semidefinite program we consider and the eigenvalue bound of Delorme and Poljak was established by Poljak and Rendl [58]. Building on work by Overton and Womersley =-=[54, 53]-=-, Alizadeh [1] has shown that eigenvalue minimization problems can in general be formulated as semidefinite programs. This is potentially quite useful, since there is an abundant literature on eigenva... |

32 | Seminumerical Algorithms." Vol. 2 of The Art of Computer Programming - Knuth - 1969 |

28 | Geometry II - Berger - 1987 |

28 |
Combinatorial properties and the complexity of a max–cut approximation
- DELORME, POLJAK
- 1993
(Show Context)
Citation Context ... by these papers, and by the paper of Alizadeh [1]. For MAX CUT, the semidefinite programming relaxation we consider is equivalent to an eigenvalue minimization problem proposed by Delorme and Poljak =-=[13, 12]-=-. An eigenvalue minimization problem consists of minimizing a decreasing sum of the eigenvaluessi of a matrix subject to equality constraints on the matrix; that is, minimizing P i m isi , wheres1s2s\... |

23 | Some applications of optimization in matrix theory
- Wolkowicz
- 1981
(Show Context)
Citation Context ...a convex cone. To some extent, semidefinite programming is very similar to linear programming; see Alizadeh [1] for a comparison. It inherits the very elegant duality theory of cone-LP (see Wolkowicz =-=[70]-=- and the exposition by Alizadeh [1]). The simplex method can be generalized to semidefinite programs (Pataki [57]). Given any ffl ? 0, semidefinite programs can be solved within an additive error of f... |

22 |
Solving the max-cut problem using eigenvalues
- Poljak, Rendl
- 1995
(Show Context)
Citation Context ...tentially quite useful, since there is an abundant literature on eigenvalue bounds for combinatorial optimization problems; see the survey paper by Mohar and Poljak [49]. As shown by Poljak and Rendl =-=[60, 59]-=- and Delorme and Poljak [14], the eigenvalue bound provides a very good bound on the maximum cut in practice. Delorme and Poljak [13, 12] study the worst-case ratio between the maximum cut and their e... |

19 |
Self-dual polytopes and the chromatic number of distance graphs on the sphere
- Lovász
- 1983
(Show Context)
Citation Context ...ional representation. We have also constructed a weighted instance on 103 vertices for which the ratio is less than .8786. These two instances are based on strongly self-dual polytopes due to Lov'asz =-=[44]-=-. A polytope P in R n is said to be strongly self-dual [44] if (i) P is inscribed in the unit sphere, (ii) P is circumscribed around the sphere with origin as center and with radius r for some 0 ! r !... |

19 |
A primal-dual interior-point method for the max-min eigenvalue problem
- Rendl, Vanderbei, et al.
- 1993
(Show Context)
Citation Context ...ptation of Ye's interior-point algorithm to semidefinite programming [1] performs O( p n(log W tot + log 1 ffl )) iterations. By exploiting the simple structure of the problem (SD) as is indicated in =-=[64]-=- (see also [68, Section 7.4]), each iteration can be implemented in O(n 3 ) time. Once an almost optimal solution to (SD) is found, one can use an incomplete Cholesky decomposition to obtain vectors v... |

17 |
A note on extreme correlation matrices
- Li, Tam
- 1994
(Show Context)
Citation Context ...tion of (SD) (i.e. which cannot be expressed as the strict convex combination of other feasible solutions) has rank at most l where l(l+1) 2sn, i.e. lsp 8n+1\Gamma1 2 ! p 2n. For related results, see =-=[41, 11, 42, 40]-=-. This means that there exists a primal optimum solution Y to (SD) of rank less than p 2n, and that the optimum vectors v i of (P ) can be embedded in R m with m ! p 2n. This result also follows from ... |

17 | New 3/4-approximation algorithms for MAX SAT - Goemans, Williamson - 1994 |

16 |
A history of non-Euclidean geometry
- Rosenfeld
- 1988
(Show Context)
Citation Context ...n to be equal to twice the area of the spherical triangle polar to the spherical triangle defined by v i , v j and v k . Stated this way, the result is a corollary to Girard's formula (1629 [20], see =-=[65]-=-) expressing the area of a spherical triangle with angles ` 1 , ` 2 and ` 3 as its excess ` 1 + ` 2 + ` 3 \Gamma . We also present a proof of the lemma from first principles. In fact, our proof parall... |

14 | A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem - Poljak, Turzik - 1982 |

13 |
Approximation and intractability results for the maximum cut problem and its variants
- Haglin, Venkatesan
- 1991
(Show Context)
Citation Context ...et S. Since 1976, a number of researchers have presented approximation algorithms for the unweighted MAX CUT problem with performance guarantees of 1 2 + 1 2m [69], 1 2 + n\Gamma1 4m [61], 1 2 + 1 2n =-=[30]-=-, and 1 2 + 1 2\Delta [33] (where n = jV j, m = jEj and \Delta denotes the maximum degree), but no progress was made in improving the constant in the performance guarantee beyond that of Sahni and Gon... |

11 |
Combinatorial optimization: some problems and trends
- Lovász
- 2000
(Show Context)
Citation Context |

11 |
How well can a graph be n-colored
- Vitányi
- 1981
(Show Context)
Citation Context ...cide which vertices are assigned to the set S. Since 1976, a number of researchers have presented approximation algorithms for the unweighted MAX CUT problem with performance guarantees of 1 2 + 1 2m =-=[69]-=-, 1 2 + n\Gamma1 4m [61], 1 2 + 1 2n [30], and 1 2 + 1 2\Delta [33] (where n = jV j, m = jEj and \Delta denotes the maximum degree), but no progress was made in improving the constant in the performan... |

10 |
A note on extreme positive definite matrices
- Christensen, Vesterstrom
- 1979
(Show Context)
Citation Context ...tion of (SD) (i.e. which cannot be expressed as the strict convex combination of other feasible solutions) has rank at most l where l(l+1) 2sn, i.e. lsp 8n+1\Gamma1 2 ! p 2n. For related results, see =-=[41, 11, 42, 40]-=-. This means that there exists a primal optimum solution Y to (SD) of rank less than p 2n, and that the optimum vectors v i of (P ) can be embedded in R m with m ! p 2n. This result also follows from ... |

10 |
The performance of an eigenvalue bound on the max-cut problem in some classes of graphs. Discrete Mathematics
- Delorme, Poljak
- 1993
(Show Context)
Citation Context ...enter of the sphere. 5 Computational Results In practice, we expect that the algorithm will perform much better than the worst-case bound of ff. Poljak and Rendl [60, 59] (see also Delorme and Poljak =-=[14]-=-) report computational results showing that the bound Z EIG is typically less than 2-5% and, in the instances they tried, never worse than 8% away from Z MC . We also performed our own computational e... |

9 |
On Cone-LP’s and Semi-definite Programs: Facial Structure, Basic Solutions and the Simplex Method
- Pataki
- 1995
(Show Context)
Citation Context ...or a comparison. It inherits the very elegant duality theory of cone-LP (see Wolkowicz [70] and the exposition by Alizadeh [1]). The simplex method can be generalized to semidefinite programs (Pataki =-=[57]-=-). Given any ffl ? 0, semidefinite programs can be solved within an additive error of ffl in polynomial time (ffl is part of the input, so the running time dependence on ffl is polynomial in log 1 ffl... |

9 |
The max-cut problem --- a survey
- Poljak, Tuza
- 1995
(Show Context)
Citation Context ...e MAX CUT problem has applications in circuit layout design and statistical physics (Barahona et al. [4]). For a comprehensive survey of the MAX CUT problem, the reader is referred to Poljak and Tuza =-=[62]-=-. Because it is unlikely that there exist efficient algorithms for NP-hard maximization problems, a typical approach to solving such a problem is to find a ae-approximation algorithm; that is, a A pre... |

8 |
Extreme points of a convex subset of the cone of positive semidefinite matrices
- Loewy
- 1980
(Show Context)
Citation Context ...tion of (SD) (i.e. which cannot be expressed as the strict convex combination of other feasible solutions) has rank at most l where l(l+1) 2sn, i.e. lsp 8n+1\Gamma1 2 ! p 2n. For related results, see =-=[41, 11, 42, 40]-=-. This means that there exists a primal optimum solution Y to (SD) of rank less than p 2n, and that the optimum vectors v i of (P ) can be embedded in R m with m ! p 2n. This result also follows from ... |

6 |
On the multiplicity of optimal eigenvalues
- Pataki
- 1994
(Show Context)
Citation Context ...hat the optimum vectors v i of (P ) can be embedded in R m with m ! p 2n. This result also follows from a more general statement about semidefinite programs due to Barvinok [5] and implicit in Pataki =-=[56]-=-: any extreme solution of a semidefinite program with k linear equalities has rank at most l where l(l+1) 2sk. 3.2 The Semidefinite Dual As mentioned in the introduction, there is an elegant duality t... |

6 | Node and edge relaxations for the max-cut problem
- POLJAK, RENDL
- 1994
(Show Context)
Citation Context ...tentially quite useful, since there is an abundant literature on eigenvalue bounds for combinatorial optimization problems; see the survey paper by Mohar and Poljak [49]. As shown by Poljak and Rendl =-=[60, 59]-=- and Delorme and Poljak [14], the eigenvalue bound provides a very good bound on the maximum cut in practice. Delorme and Poljak [13, 12] study the worst-case ratio between the maximum cut and their e... |

6 | Extremal correlation matrices - Grone, Pierce, et al. - 1990 |

5 |
Extremal correlation matrices. Linear Algebra and its Applications
- Grone, Pierce, et al.
- 1990
(Show Context)
Citation Context ...ax 1 2 X i!j w ij (1 \Gamma y ij ) (SD) subject to: y ii = 1 8i 2 V Y symmetric positive semidefinite 9 where Y = (y ij ). The feasible solutions to (SD) are often referred to as correlation matrices =-=[24]-=-. Strictly speaking, we cannot solve (SD) to optimality in polynomial time; the optimal value Z P might in fact be irrational. However, using an algorithm for semidefinite programming, one can obtain,... |

5 |
A combinatorial design approach to Max Cut. Random Structures and Algorithms
- Hofmeister, Lefmann
- 1996
(Show Context)
Citation Context ... of researchers have presented approximation algorithms for the unweighted MAX CUT problem with performance guarantees of 1 2 + 1 2m [69], 1 2 + n\Gamma1 4m [61], 1 2 + 1 2n [30], and 1 2 + 1 2\Delta =-=[33]-=- (where n = jV j, m = jEj and \Delta denotes the maximum degree), but no progress was made in improving the constant in the performance guarantee beyond that of Sahni and Gonzales's straightforward al... |

5 |
Matrix cones, projection representations, and stable set polyhedra
- Lov'asz, Schrijver
- 1990
(Show Context)
Citation Context |

5 | Algorithms for cone-optimization problems and semi-definite programming,” Graduate - Pataki - 1994 |

3 |
Eigenvalue methods in combinatorial optimization
- Mohar, Poljak
- 1993
(Show Context)
Citation Context ...s semidefinite programs. This is potentially quite useful, since there is an abundant literature on eigenvalue bounds for combinatorial optimization problems; see the survey paper by Mohar and Poljak =-=[49]-=-. As shown by Poljak and Rendl [60, 59] and Delorme and Poljak [14], the eigenvalue bound provides a very good bound on the maximum cut in practice. Delorme and Poljak [13, 12] study the worst-case ra... |

2 | Approximating maximum 2-CNF satisfiability
- Haglin
- 1992
(Show Context)
Citation Context ...hat the existence of a c-approximation 15 algorithm implies that P = NP [2]. Bellare, Goldreich, and Sudan [6] have shown that a 95/96-approximation algorithm for MAX 2SAT would imply P = NP . Haglin =-=[28, 29]-=- has shown that any ae-approximation algorithm for MAX RES CUT can be translated into a aeapproximation algorithm for MAX 2SAT, but we will show a direct algorithm here. Haglin's observation together ... |

2 |
Finding the maximal cut in a graph. Engineering Cybernetics
- Orlova, Dorfman
- 1972
(Show Context)
Citation Context ...ete even if the problem is unweighted; that is, if w ij = 1 for all (i; j) 2 E [19]. The MAX CUT problem is solvable in polynomial time for some special classes of graphs (e.g. if the graph is planar =-=[52, 27]-=-). Besides its theoretical importance, the MAX CUT problem has applications in circuit layout design and statistical physics (Barahona et al. [4]). For a comprehensive survey of the MAX CUT problem, t... |

2 | Computational experiments with node and edge relaxations of the max-cut problem - Poljak, Rendl - 1994 |

1 |
De mensura angulorum solidorum
- Euler
- 1781
(Show Context)
Citation Context ... a spherical triangle with angles ` 1 , ` 2 and ` 3 as its excess ` 1 + ` 2 + ` 3 \Gamma . We also present a proof of the lemma from first principles. In fact, our proof parallels Euler's proof (1781 =-=[15]-=-, see [65]) of Girard's formula. We define the following events: A : sgn(v i \Delta r) = sgn(v j \Delta r) = sgn(v k \Delta r) B i : sgn(v i \Delta r) 6= sgn(v j \Delta r) = sgn(v k \Delta r) C i : sg... |

1 |
De la mesure de la superficie des triangles et polygones sph'eriques. Appendix to "Invention nouvelle en l'alg`ebre
- Girard
- 1629
(Show Context)
Citation Context ...can be seen to be equal to twice the area of the spherical triangle polar to the spherical triangle defined by v i , v j and v k . Stated this way, the result is a corollary to Girard's formula (1629 =-=[20]-=-, see [65]) expressing the area of a spherical triangle with angles ` 1 , ` 2 and ` 3 as its excess ` 1 + ` 2 + ` 3 \Gamma . We also present a proof of the lemma from first principles. In fact, our pr... |

1 |
A highly parallel implementation of the Goemans/Williamson algorithm to approximate MaxCut
- Homer, Peinado
- 1994
(Show Context)
Citation Context ... proven for these Euclidean instances. However, the instance defined by a unit length equilateral triangle has a maximum cut value of 2, but Z P = 9 4 , for a ratio of 8 9 = 0:8889. Homer and Peinado =-=[34]-=- have implemented our algorithm on a CM-5, and have shown that it produces optimal or very nearly optimal solutions to a number of MAX CUT instances derived from via minimization problems. These insta... |

1 |
On the facial structure of the set of correlation matrices. Unpublished manuscript
- Laurent, Poljak
- 1995
(Show Context)
Citation Context |

1 |
Correctly derandomizing Goemans and Williamson's Max CUT algorithm. Unpublished manuscript
- Mahajan, Ramesh
- 1995
(Show Context)
Citation Context ...inistic versions of our approximation algorithm with the same performance guarantees. However, the method given had a subtle error, as was pointed out to us by several researchers. Mahajan and Ramesh =-=[48]-=- document the error and propose their own derandomization scheme for our algorithms. The paper is structured as follows. We present the randomized algorithm for MAX CUT in Section 1, and its analysis ... |

1 | An application of connbinatorial optimization to statistical physics and circuit layout design - BARAHONA, GROTSCHEL, et al. - 1988 |

1 | Free bits, PCP and non-approximabili~y— Tovvardstight results - BELLARE, GOLDREICH, et al. - 1995 |

1 | Geomety 11 - BERGIER - 1987 |

1 | Graph Theo~ with Applications - BONDY, MURTY - 1976 |

1 | Approximation and intractability results for the maximum cut problem and its variants - J, ANDVENKATESAN - 1991 |

1 | Approximate graph coloring by semidefinite programming - nRGER, mx, et al. - 1994 |

1 | The Theoiy of Matrices - Orlando, ANDPOUW, et al. - 1985 |

1 | On the Shannon capacity of a graph - SZ - 1979 |

1 | Combinatorial optimization: Some problems and trends - SZ - 1992 |

1 | Cones of matrices and setfunctions, and O–1 optimization - Loviisz, SCHRIJVER - 1990 |

1 | Finding the maximal cut in a graph - OVERTON, WOMERSLEY - 1972 |

1 | Optimization, approximation, and complexhy classes.J - H, YANNAKAKJS, et al. - 1991 |

1 | Algorithms for Maximum Cut and Satisfiability Problems 1145 - PATAKI - 1995 |

1 | Node and edge relaxations of the max-cut problem - POUAK, ANDRENDL - 1994 |

1 | Maximum cuts and largest bipartite subgraphs - ML, ANDTUZA - 1995 |

1 | TSPLIB—A traveling salesman problem library - T - 1991 |

1 | A histo~ of non-Euclidean geometry - ROSENIFELD - 1988 |

1 | How well can a graph be n-colored? Disc - VITANYI - 1981 |

1 | Some applications of optimization in matrix theory - WOLKC, H - 1981 |