C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62:557--574, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....implies the following upper bound on mc(G) which was first observed by Mohar and Poljak [M P1] # max (G) 22) Similarly to the bipartition width problem, the bound (22) can be further improved using correction functions. The following eigenvalue bound was introduced by Delorme and Poljak [D P1]. Proposition 3.6 Let G be a weighted graph of order n. Then # max (L(G) diag(c) 23) where the minimum is taken over all correction functions c . Proof. The proof consists of similar steps as the proof of Proposition 3.3. Choose V (G) such that e(S, S) mc(G) and let g ....

C. Delorme, S. Poljak, Laplacian eigenvalues and the maximum cut problem, Math. Programming 62 (1993) 557--574. 47

.... b T y and H(y) P m k=1 y k A k Gamma C. Example 2. SDP Relaxations of Binary Combinatorial Optimization Problems In SDP relaxations of binary combinatorial optimization problems, the binary constraints x 2 i = 1; i = 1; 2; n; are relaxed into X ii = 1; i = 1; 2; n; see [19,13,5,16] for the evolution of such relaxations) resulting in primal linear SDP problems in the form of (4) with I = D and B I = I, the identity matrix. The dual form of these SDP relaxations are special instances of (5) In particular, when m = 0, we obtain the MAXCUT SDP relaxation that forms the ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62:557-574, 1993. Burer, Monteiro and Zhang: Solving Semidefinite Programs via Nonlinear Programming

....general Max Cut Problem has de ed any ecient solution so far, and indeed, it was proved to be NP hard [10] even in the case when all edge weights are equal to 1. In spite of that, many attempts have been made to tackle the problem with approximation and randomized algorithms. Delorme and Poljak [6] and Poljak and Rendl [20] solved a relaxation of the problem using eigenvalues. A similar approach based on semide nite programming was developed by Goemans and Williamson who presented a randomized algorithm in [12] with a performance guarantee of 0:878. Polynomial time methods to nd the exact ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Prog., 62:557-574, 1993.

....variable is replaced by a matrix valued continuous variable, resulting in a convex optimization problem called a semidefinite program (SDP) that can be solved to a prescribed accuracy in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [10, 23, 24, 26, 27]. Based on solving the SDP relaxation, Goemans and Williamson [18] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time ....

C. Delorme and S. Poljak. Laplacian Eigenvalues and the Maximum Cut Problem. Mathematical Programming, 62:557--574, 1993.

....and its dual is precisely the SDP problem above. Again, these dual problems satisfy the conditions of the next section guaranteeing strong duality, so either provides a relaxation of the original max cut problem. These bounds on the value of a maximum weight cut were obtained by Delorme and Poljak [14]. Since we have a relaxation, the optimal value of the SDP problem provides an upper bound on the value of the max cut. But in this case, we can also use the solution of the primal problem to generate a provably good cut, as was shown in a beautiful contribution of Goemans and Williamson [22] see ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Programming, 62:557--574, 1993.

....plane algorithms, interior point methods. AMS subject classifications (MSC 1991) 90C25, 90C09 1 Introduction Pioneered by the work of Lov asz on the Shannon capacity of graphs [27] the interest in semidefinite relaxations of combinatorial optimization problems has been steadily increasing [16, 28, 7, 34]. With the development of interior point algorithms [23, 32, 1, 37, 21, 25, 2, 33] practical methods for computing these relaxations became available. This encouraged research in the field and, as a consequence, several new approximation results based on semidefinite relaxations appeared within ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62:557--547, 1993.

....Houston, Texas 77005, USA. This author was supported in part by DOE Grant DE FG03 97ER25331, DOE LANL Contract 03891 99 23 and NSF Grant DMS 9973339. Email: zhang caam.rice.edu) 1 in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [9, 17, 18, 20, 21]. Based on the SDP relaxation, Goemans and Williamson [14] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time approximation ....

C. Delorme and S. Poljak. Laplacian Eigenvalues and the Maximum Cut Problem. Mathematical Programming, 62:557-574, 1993.

....Houston, Texas 77005, USA. This author was supported in part by DOE Grant DE FG03 97ER25331, DOE LANL Contract 03891 99 23 and NSF Grant DMS 9973339. Email: zhang caam.rice.edu) 1 in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [9, 17, 18, 20, 21]. Based on the SDP relaxation, Goemans and Williamson [14] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time approximation ....

C. Delorme and S. Poljak. Laplacian Eigenvalues and the Maximum Cut Problem. Mathematical Programming, 62:557--574, 1993.

....The general Max Cut problem has defied efficient solution so far, and indeed, it was proved to be NP hard [GJ] even in the case when all edge weights are equal to 1 or Gamma1. In spite of that, many attempts have been made to tackle the problem with approximation and randomized algorithms ([DP], PR] GW] Polynomial time methods for toroidal square lattices were suggested in the early 60 s by Kasteleyn [Kast, Kast1] and Kac and Ward [KW] Kac and Ward tried to calculate the partition function as a determinant of a 4n Theta 4n matrix over complex numbers and even though their ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Prog., 62:557--574, 1993.

....general MAX CUT problem has defied efficient solution so far, and indeed, it was proved to be NP hard [9] even in the case when all edge weights are equal to 1. In spite of that, many attempts have been made to tackle the problem with approximation and randomized algorithms. Delorme and Poljak [6] and Poljak and Rendl [18] solved a relaxation of the problem using eigenvalues. A similar approach based on semidefinite programming was developed by Goemans and Williamson who presented a randomized algorithm in [11] with a performance guarantee of 0:878. Polynomial time methods to find the ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Prog., 62:557--574, 1993.

....continuous variable, resulting in a convex optimization problem called a SDP problem. Since a SDP problem is solvable in polynomial time, one can obtain a bound to the original problem in polynomial time. Some early ideas related to such a relaxation can be found in a number of works, including [9, 18, 19, 21, 22]. Based on the SDP relaxation, Goemans and Williamson [14] proposed a randomized algorithm for the Max Cut problem and established the celebrated 0.878 performance guarantee. Since then, SDP relaxation has become a powerful and popular theoretical tool for devising polynomial time approximation ....

C. Delorme and S. Poljak. Laplacian Eigenvalues and the Maximum Cut Problem. Mathematical Programming, 62:557--574, 1993.

....entries u 1 ; u n and max (L(G) diag(u) is the largest eigenvalue of the matrix L(G) diag(u) Set (G) max( 1 2 Trace(AY ) 1 2 J Gamma Y is positive semidefinite and Y ij = 0 for 1 i n) where J is the n Theta n matrix with all entries equal to 1. Then, mc(G) G) [27]) and mc(G) G) 60] In fact, by general duality theory, these two bounds coincide, i.e. G) G) 55] It is easy to see that (G) max( X ij2E(G) x satisfies the inequalities (36) for all b 2 Z n ) X 1i jn b i b j x ij oe 2 4 (36) The inequalities (36) are clearly ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, submitted. 98 M. Deza, V.P. Grishukhin and M. Laurent

.... problem is to solve the graph partitioning problem with k = 2, m 1 = i, and m 2 = n Gamma i for all i = 1; Delta Delta Delta bn=2c (notice that in graph partitioning problem max and min characterizations are essentially equivalent by simply changing the weights w i with P w j Gamma w i ) In [DP90, PR91] the following SDP bound is proposed: minf n 4 1 (A Diag(x) 1 T x = ag MC(G) 76) where MC(G) is the size of maximum cut in G. 5.76) is equivalent to primal dual pair: min z (1=n)1 T x s.t. zI Gamma Diag(x) A max A ffl Y s.t. Y ii = 1=n Y 0 (77) 33 and may be solved by ....

C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Technical Report 599, Universi'e de Paris-sud, Centre d'Orsay, 1990.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Programming, 62:557--574, 1993.

....cannot be found efficiently unless NP = P , various approximating procedures have been proposed in the literature. An approximation of the max cut based on the minimization of the maximum eigenvalue of the Laplacian matrix with respect to diagonal changes, has been introduced and studied in [3, 4, 5]. The computational experiments of [18] show that the eigenvalue bound provides a good approximation of the max cut, since the relative error typically ranges between 1 5 . It has been shown in [17] that the dual formulation of the eigenvalue bound is the optimization problem max c t x x 2 J ....

....that Y 62 fD Gamma M j M 2 PSD n ; D 2 DIAG n ; Tr(D) 1g. Then, for each D 2 DIAG n with Tr(D) 1, the matrix D Gamma Y is not positive semidefinite, i.e. min (D Gamma Y ) 0. Therefore, we have that max( min (D Gamma Y ) j D 2 DIAG n ; Tr(D) 1) 0. The following result is shown in [3]. Let D 0 be the diagonal matrix with trace one for which the above maximization problem attains its optimum and set 0 = min (D 0 Gamma Y ) 0. Then, there exists a set of vectors v 1 ; v k which are eigenvectors of D 0 Gamma Y for the eigenvalue 0 and such that all diagonal entries ....

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, to appear.

....w ij (1 Gamma x i x j ) x 2 F; where x i = 1 if i 2 I and 1 otherwise. This is an instance of (P) with a pure quadratic objective function. The corresponding matrix Q has components q ij = Gammaw ij ; with 0 diagonal. Alternatively, Q can be taken as the Laplacian matrix of the graph, see e.g. [6]. The resulting eigenvalue bound and the equivalent semidefinite bound have been recently used in numerical studies and found to do exceptionally well, e.g. 13] Moreover, the semidefinite bound has been studied in [6] and shown to have a particular good performance index, see [10] RECIPE FOR ....

.... Alternatively, Q can be taken as the Laplacian matrix of the graph, see e.g. 6] The resulting eigenvalue bound and the equivalent semidefinite bound have been recently used in numerical studies and found to do exceptionally well, e.g. 13] Moreover, the semidefinite bound has been studied in [6] and shown to have a particular good performance index, see [10] RECIPE FOR SDP RELAXATION FOR (0,1) QUADRATIC PROGRAMMING 277 Other instances that we consider include: graph partitioning and max clique problems. 1.2. HISTORICAL BACKGROUND Quadratic bounds using a Lagrangian relaxation have ....

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Delorme, C. and Poljak, S., Laplacian eigenvalues and the maximum cut problem. Math. Programming, 62(3):557--574, 1993.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62:557--574, 1993.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Prog., 62:557--574, 1993.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62:557--574, 1993.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Math. Programming, 62(3, Ser. A):557--574, 1993.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62:557--574, 1993.

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C. Delorme and S. Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62(3, Ser. A):557--574, 1993.

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C. Delorme and S. Poljak, "Laplacian Eigenvalues and the Maximum Cut Problem," Math. Programming, vol. 62, pp. 557574, 1993.

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C. DELORME and S. POLJAK. Laplacian eigenvalues and the maximum cut problem. Math. Programming, 62(3):557--574, 1993.

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C. Delorme, S. Poljak, Laplacian eigenvalues and the maximum cut problem, Math. Programming 62 (1993) 557--574. 47

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