S. Poljak and F. Rendl. Node and edge relaxations of the max-cut problem. Computing, 52:123--137, 1994. Also appears as Report 266, Technische Universitat Graz.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....eigenvalue minimization problems can in general be formulated as 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 ratio between the maximum cut and their eigenvalue bound. The worst instance they are aware of is the 5 cycle for which the ratio is 32 ....

....is less than :8796 in the unweighted case. The convex hull of the optimum vectors is depicted on the right; the circle represents the center 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 experiments, in which the cuts computed by the algorithm ....

S. Poljak and F. Rendl. Node and edge relaxations of the max-cut problem. Computing, 52:123--137, 1994. Also appears as Report 266, Technische Universitat Graz.

....C Z Gamma Diag(y) 0 Z 0: For non negatively weighted graphs a celebrated result of Goemans and Williamson [8] says, that there is always a cut within :878 of the optimal value of the relaxation. One of the first attempts to approximate (DMC) using eigenvalue optimization is contained in [39]. The authors use the Bundle code of Schramm and Zowe [42] with a limited number of bundle iterations, and so do not solve (DMC) exactly. So far the only practical algorithms for computing the optimal value were primal dual interior point algorithms. However these are not able to exploit the ....

S. Poljak and F. Rendl, Node and edge relaxations of the max-cut problem, Computing, 52 (1994), pp. 123--137.

....comes from (3) Therefore the largest eigenvalue of the Laplacian provides an upper bound on the weight of a maximum cut. A further improvement is proposed in [12] using (4) to obtain 4mc(G) G) min u t e=0 n max (L diag(u) 5) Computational experiments are contained in [37] and in [35], where this bound is applied to huge graphs (with up to 50,000 vertices and several million edges) Using some of the combinatorial properties of (G) investigated in [11] the paper [37] provides the first results of this bound in a Branch and Bound setting. In [36] it is shown, that the ....

S. POLJAK and F. RENDL. Node and Edge Relaxations of the Max-Cut Problem. Computing, 52:123--137, 1994.

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S. Poljak and F. Rendl. Node and edge relaxations of the max-cut problem. Computing, 52:123--137, 1994. Also appears as Report 266, Technische Universitat Graz.

No context found.

S. Poljak and F. Rendl. Node and edge relaxations of the max-cut problem. Computing, 52:123--137, 1994. Also appears as Report 266, Technische Universitat Graz.

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