## Semidefinite Programming and Combinatorial Optimization (1998)

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Venue: | DOC. MATH. J. DMV |

Citations: | 96 - 1 self |

### BibTeX

@MISC{Goemans98semidefiniteprogramming,

author = {Michel X. Goemans},

title = { Semidefinite Programming and Combinatorial Optimization},

year = {1998}

}

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### Abstract

We describe a few applications of semide nite programming in combinatorial optimization.

### Citations

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1130 | Algorithmic Graph Theory and Perfect Graphs - Golumbic - 1984 |

934 | Improved approximation algorithms for maximum cuts and satisfiability problems using semidefinite programming. Journal of the Association for Computing Machinery, 42:1115–1145
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...ithm. The simplest and most illustrative example where semidefinite programming helps in the design of approximation algorithms is the maximum cut problem, as was discovered by Goemans and Williamson =-=[21]-=-. Let us assume that we are given a graph G = (V; E) with nonnegative edge weights w e for e 2 E and we would like to find a cut of maximum weight. It is easy to obtain a 0:5-approximation algorithm b... |

766 | Semidefinite programming - Vandenberghe, Boyd - 1996 |

640 | Some optimal inapproximability results
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(Show Context)
Citation Context ... value [39]. Thus if we compare the solution obtained by any algorithm to this linear programming relaxation we cannot expect a performance guarantee ff better than 0:5. Furthermore, recently, Hastad =-=[25]-=-, based on a long series of developments in the area of probabilistically checkable proofs, has shown that it is NP-hard to approximate the maximum cut problem within 16=17 + ffl = 0:94117 \Delta \Del... |

470 | Interior point methods in semidefinite programming with applications in combinatorial optimization
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- 1995
(Show Context)
Citation Context ...hey propose the following relaxation, w cutsmax u T e=0 min Y T Y =I k Trace 1 2 Y T (L + Diag (u))Y = max u T e=0 1 2 X jsj (L + Diag (u)) =: wDH : It is a simple exercise using duality for SDP, see =-=[1]-=-, to rewrite this lower bound as wDH = min 1 2 L ffl X such that mIsXs0; diag(X) = e; corresponding to the constraints 2., 4. and 5 for F k . Rendl and Wolkowicz [40] improved this model to get w cuts... |

450 | The geometry of graphs and some of its algorithmic applications - Linial, London, et al. - 1995 |

403 |
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Citation Context ...ey idea of this approach is to use the concept of algebraic connectivity to enforce connectivity. The algebraic connectivity ff(A) of a graph given by its adjacency matrix A was introduced by Fiedler =-=[17]-=-, and is defined to be the second-smallest (Laplacian) eigenvalue of L(A), ff(A) :=s2 (L(A)): Fiedler [17] shows that ff(A) :=s2 (L(A), behaves in many ways quite similar to the edge-connectivity (A) ... |

364 | The Theory of Matrices - Lancaster, Tismenetsky - 1985 |

362 | The Ellipsoid Method and its Consequences in Combinatorial Optimization - Grötschel, Lovász, et al. - 1981 |

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329 | Eigevalues and Expanders - Alon - 1986 |

277 | On Lipschitz embedding of finite metric spaces in Hilbert space - Bourgain - 1985 |

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257 | Approximate counting, uniform generation and rapidly mixing Markov chains - Sinclair, Jerrum - 1989 |

239 | An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms - LEIGHTON, RAO - 1988 |

228 | An algebraic approach to the association schemes of coding theory. Philzps Res. Rep. Suppl., (10):vi+97, 1973. P. Delsarte. The association schemes of coding theory - Delsarte - 1974 |

217 | A lift-and-project cutting plane algorithm for mixed 0-1 programs - Balas, Ceria, et al. - 1993 |

185 | A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems - Sherali, Adams - 1990 |

178 | Approximate graph coloring by semidefinite programming
- Karger, Motwani, et al.
- 1998
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Citation Context ...tic problems, including the maximum directed cut problem [21, 15], the maximum k-cut problem and bisection problems [18, 43], maximum satisfiability problems (see [21, 15, 32, 47, 4]), graph coloring =-=[29]-=-, machine scheduling problems [41], and more general quadratic problems (see Section 13.2 of this Handbook for details). Although the random hyperplane technique is the only currently known method for... |

160 | Improved approximation algorithms for MAX k-CUT
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Citation Context ...hnique has been applied to several problems that can be modelled naturally as quadratic problems, including the maximum directed cut problem [21, 15], the maximum k-cut problem and bisection problems =-=[18, 43]-=-, maximum satisfiability problems (see [21, 15, 32, 47, 4]), graph coloring [29], machine scheduling problems [41], and more general quadratic problems (see Section 13.2 of this Handbook for details).... |

143 |
Lower bounds for the partitioning of graphs
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(Show Context)
Citation Context ... investigated mostly for k = 2, and use additional combinatorial cutting planes, see [8, 6]. Surprisingly enough, the purely linear relaxation was not the first model investigated. Donath and Hoffman =-=[11]-=- use the following well-known characterization of Fan [13] for the sum of the k smallest eigenvalues of a symmetric matrix A, k X j=1sj (A) = min Y T Y =I k Trace Y T AY: They propose the following re... |

138 | A spectral bundle method for semidefinite programming
- Helmberg, Rendl
(Show Context)
Citation Context ...isfied both because of computer memory and computation time limitations. Therefore one is interested in finding the most important constraints quickly. To give a flavor of how this works, we refer to =-=[27]-=- for computational results in the case of the Max-Cut problem, and to [30] for general graph bisection. In Tables 1 and 2 we summarize some more recent computational results on real world data for EQP... |

130 | Approximating the value of two prover proof systems, with applications to MAX - Feige, Goemans - 1995 |

124 | An O(log k) approximate min-cut max-flow theorem and approximation algorithm - Aumann, Rabani - 1998 |

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114 |
Semidefinite relaxation and nonconvex quadratic optimization
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Citation Context ...gs to a cone K, we can bound fi by imposing that 2sarcsin[X] \Gamma fiX 2 K whenever Ks0; diag(X) = e. When the cone K is the positive semidefinite cone, fi can be chosen to be 2 as shown by Nesterov =-=[37]-=-. The random hyperplane technique has been applied to several problems that can be modelled naturally as quadratic problems, including the maximum directed cut problem [21, 15], the maximum k-cut prob... |

108 | A 7/8-approximation algorithm for MAX 3SAT
- Karloff, Zwick
- 1997
(Show Context)
Citation Context ...t can be modelled naturally as quadratic problems, including the maximum directed cut problem [21, 15], the maximum k-cut problem and bisection problems [18, 43], maximum satisfiability problems (see =-=[21, 15, 32, 47, 4]-=-), graph coloring [29], machine scheduling problems [41], and more general quadratic problems (see Section 13.2 of this Handbook for details). Although the random hyperplane technique is the only curr... |

108 | On the cut polytope - Barahona, Mahjoub - 1986 |

101 | Complementarity and nondegeneracy in semidefinite programming - Alizadeh, Haeberly, et al. - 1997 |

99 | New upper bounds on the rate of a code via the Delsarte–McWilliams inequalities - McEliece, Rodemich, et al. - 1977 |

88 | Eigenvalue optimization - Lewis, Overton - 1996 |

77 | A characterization of perfect graphs - Lovasz - 1972 |

73 |
The Quadratic Assignment Problem: Theory and Algorithms
- Cela
- 1998
(Show Context)
Citation Context ... usually formulated in the following way. For given symmetric matrices A; B and arbitrary matrix C, all of order n, find a permutation matrix X = (x ij ) minimizing Trace ((AXB + C)X T ): We refer to =-=[7]-=- for a comprehensive summary on QAP. A direct attempt to model this quadratic function using SDP leads immediately to matrices of order n 2 . On the other hand, the HoffmanWielandt inequality provides... |

70 | Semidefinite programming relaxations for the quadratic assignment problem
- Zhao, Karisch, et al.
- 1998
(Show Context)
Citation Context ...n matrices. It is possible, though, to use the left hand side model in Theorem 2.5 to warm-start it. Further refinements and variations of this model along with computational results are contained in =-=[45]-=-. 3 Computational aspects It is well known that SDP can be solved in polynomial time to some fixed precision. In practice, interior point methods have turned out most efficient for SDP whenever dense ... |

67 | A 1, isoperimetric inequalities for graphs and superconcentrators - Alon, Milman - 1985 |

67 | Clique is hard to approximate within n 1\Gammaffl - Hastad - 1999 |

67 | Expressing combinatorial optimization problems by linear programs - Yannakakis - 1991 |

63 |
On a theorem of weyl concerning eigenvalues of linear transformations i
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(Show Context)
Citation Context ...torial cutting planes, see [8, 6]. Surprisingly enough, the purely linear relaxation was not the first model investigated. Donath and Hoffman [11] use the following well-known characterization of Fan =-=[13]-=- for the sum of the k smallest eigenvalues of a symmetric matrix A, k X j=1sj (A) = min Y T Y =I k Trace Y T AY: They propose the following relaxation, w cutsmax u T e=0 min Y T Y =I k Trace 1 2 Y T (... |

60 | Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to Max-Cut and other problems
- Zwick
- 1999
(Show Context)
Citation Context ...)]s0:87856z sdp since arccos(x)s0:87856 1\Gammax 2 for \Gamma1sxs1. Karloff [31] has shown that the ratio E[w(ffi(S))]=z sdp can be arbitrarily close to 0:87856. Goemans and Williamson [21] and Zwick =-=[47]-=- have shown that an improvement over the 0:87856 ratio for the performance guarantee can be given as a function of ae = z sdp = P (i;j)2E w ij , unless ae is 0:84458 \Delta \Delta \Delta . Furthermore... |

59 |
Eigenvalues and graph bisection: an average case analysis
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- 1987
(Show Context)
Citation Context ... graph bisection problem, or 2-EQP. This problem asks to partition the vertices of a graph into two sets of equal cardinality so as to minimize the total weight of edges cut by the partition. Boppana =-=[5]-=- shows that a lower bound to the equicut problem is given by n 4s2 (L), if L denotes the weighted Laplacian. Modeling bisections by (1; \Gamma1) vectors x, the equicut problem can be written as min 1 ... |

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

49 | A comparaison of the Delsarte and Lovász bound - Schrijver - 1979 |

45 | Lagrangian relaxation of quadratic matrix constraints
- Anstreicher, Wolkowicz
- 2000
(Show Context)
Citation Context ...of A and B is minimized. Since permutation matrices are orthogonal, this result can be used to bound the quadratic part of the cost function of QAP. It was recently shown by Anstreicher and Wolkowicz =-=[3]-=- that in fact the minimization in (2.6) can equivalently be expressed as an SDP. Theorem 2.5 [3] minfTrace (AXBX T ) : X T X = Ig = max fTrace S + Trace T : (B\Omega A) \Gamma (I\Omega S) \Gamma (T\Om... |

45 | Problems of distance geometry and convex properties of quadratic maps - Barvinok - 1995 |

41 | Randomized graph products, chromatic numbers, and the Lovász ϑ-function
- Feige
- 1997
(Show Context)
Citation Context ...solve the weighted problem, or find the chromatic number or the largest clique. Although #(G) = ff(G) for perfect graphs, #(G) can provide a fairly poor upper bound on ff(G) for general graphs. Feige =-=[14]-=- has shown the existence of graphs for which #(G)=ff(G)s\Omega\Gamma n 1\Gammaffl ) for any ffl ? 0. See [19] for further details and additional references on the quality of #(G). 2.3 Traveling salesm... |

41 | A Projection Technique for Partitioning the Nodes of Graph
- Rendl, Wolkowicz
- 1990
(Show Context)
Citation Context ...xercise using duality for SDP, see [1], to rewrite this lower bound as wDH = min 1 2 L ffl X such that mIsXs0; diag(X) = e; corresponding to the constraints 2., 4. and 5 for F k . Rendl and Wolkowicz =-=[40]-=- improved this model to get w cutsmax u T e=0 min Y T Y =I k ;Y e=me;Y T e=e 1 2 Trace Y T (L + Diag (u))Y =: wRW : Using duality again, it can be shown, see [30] that wRW = min 1 2 L ffl X such that ... |

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