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

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Venue: | Journal of the ACM |

Citations: | 937 - 14 self |

### BibTeX

@ARTICLE{Goemans95improvedapproximation,

author = {M. X. Goemans and D.P. Williamson},

title = {Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming},

journal = {Journal of the ACM},

year = {1995},

volume = {42},

pages = {1115--1145}

}

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### Abstract

We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (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 ...

### Citations

1430 |
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... |

1258 | Graph Theory with Applications - Bondy, Murty - 1976 |

1139 |
Geometric Algorithms and Combinatorial Optimizations
- Grotschel, Lovász, 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... |

769 | 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... |

717 | Proof Verification and the Hardness of Approximation Problems
- Arora, Lund, et al.
- 1998
(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... |

572 |
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... |

473 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(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... |

376 |
Some simplified NP-complete graph problems
- Garey, Johnson, et al.
- 1976
(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... |

364 |
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,... |

362 |
The ellipsoid method and its consequences in combinatorial optimization
- Grötschel, Lovász, et al.
- 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]... |

338 |
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... |

278 |
P-complete approximation problems
- Sahni, Gonzales
- 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... |

262 | 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... |

259 |
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... |

248 | Interior-point polynomial methods in convex programming, ser - Nesterov, Nemirovsky - 1994 |

206 | Free bits, PCPs and non-approximability — 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... |

179 | Approximate graph coloring by semidefinite programming
- Karger, Motwani, et al.
- 1998
(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... |

161 | Improved Approximation Algorithms for Max k-cut and Max Bisection.", Integer Programming and Combinatorial Optimization
- Frieze, Jerrum
- 1995
(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... |

144 |
Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, third edition
- 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... |

132 | Approximating the value of two prover 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... |

120 |
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... |

120 |
A new algorithm for minimizing convex functions over convex sets
- Vaidya
- 1996
(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... |

108 |
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... |

94 | 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... |

72 |
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... |

70 |
878-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... |

69 | New 3/4-approximation algorithms for the maximum satisfiabihty problem
- Goemans, Wiffiamson
- 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 ... |

62 | Optimality conditions and duality theory for minimizing sums of the Lqest eigenvalues of symmetric matrices. Mathemicd Programming
- Overton, Wamersley
- 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... |

61 |
A linear time approximation algorithm for the weighted vertex cover problem
- Bar-Yehuda, Even
(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... |

59 |
An application of combinatorial optimization to statistical physics and circuit layout design," Operations Resear«h 36
- Barahona, Grötschel, 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 ... |

56 |
Laplacian eigenvalues and the maximum cut problem
- Delorme, Poljak
(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\... |

46 |
Problems of distance geometry and convex properties of quadartic 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... |

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

41 | Eigenvalues in combinatorial optimization - MOHAR, POIAAK - 1993 |

39 | 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 | 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 ... |

33 |
Self-concordant functions and polynomial time methods in convex programming
- Nesterov, Nemirovski
- 1990
(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. ... |

31 |
On the sum of the largest eigenvalues of a symmetric matrix
- Overton, R
(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... |

27 |
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 | Geometry II - Berger - 1977 |

22 | Seminumerical Algorithms, vol. 2 of The Art of Computer Programming - Knuth - 1998 |

20 |
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 |
A primal-dual interior point method for the max-min eigenvalue problem
- Helmberg, Rendl, et al.
(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 | New 3/4-approximation algorithms for MAX SAT - Goemans, Williamson |

13 |
Combinatorial optimization: some problems and trends
- Lovász
- 1992
(Show Context)
Citation Context |

13 | Turzik D.: A polynomial algorithm for constructing a large bipartite subgraph with an application to satisfiability problem - Poljak - 1982 |

12 |
S.: 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 |
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 !... |

11 |
How well can a graph be n-colored
- Vit'anyi
- 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... |