H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm ? SIAM J. on Computing, 29(1):336--350, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... for which the optimal value of therelaxation is equal to the size of the maximum cut an for which there isan optimalsolution of therelaxation in which theanxI between every two vectors that correspon to verticesin the graph that areconI)1fiS byan edge is exactly, or very close to, # 0 Karlo# [Kar99] was the first toconqS3qx graphs that satisfy theseconxS)fiI an therefore show that the localanxI)S of the MAX CUT approximation algorithm of Goeman an Williamson isinISq tight. Karlo# s result was simplified byAlon an Sudakov [AS00] Goeman an Williamson [GW95] give a better lower boun on the ....

....the size (or weight) of the maximum cut an then umber of edges (or total weight of the edges) of G. Note that 1 2 A# 1. It is shown in [GW95] that if A t 0 0.84458, where t 0 0 t#1 h(t) t an h(t) arccos(1 2t) #,then performan) ratio of the MAX CUT algorithm is at least #(A) h(A) A #. Kar99]an Alon an Sudakov [AS00] show that this lower boun isagain tight for every t 0 1. What happen on graphs with 1 2 A t 0 Goeman an Williamson [GW95] can showonS that the performan1 ratio of their algorithmon such graphs is at least #. Zwick [Zwi99]presen ts a modification of the algorithm ....

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H. Karloff, How good is the Goemans--Williamson MAX CUT algorithm?Spd J. Comput., 29 (1999), pp. 336--350.

....[11] introduced the idea of using methods from Semidefinite Programming to approximate the solution with guaranteed bounds on the error better than the naive value of 3 4 . Semidefinite programming methods involve a lot of machinery, and in practice, their efficacy is sometimes questioned [14]. Efficient Solution Rather than using semidefinite approaches, we will resort to spectral partitioning [27] for computational efficiency reasons. In doing so, we exploit particularly two additional facts that hold in our situation: 1. Rather than being a general graph, we have an instance which ....

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In Proc. of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 427--434, Philadelphia, Pennsylvania, 22--24 May 1996.

....ffl. We now turn to the approximation ratio for the GW algorithm. Recall that we may assume that alg exp. Using also the fact that opt sdp, the analysis of Goemans and Williamson implies that the approximation ratio is at least ff. But is it better than ff This question was studied by Karloff [13]. For every ffl 0, he constructed a graph that has the following properties: 1. sdp = opt. 2. The graph has an optimal embedding on the sphere for which exp (ff ffl)sdp. Thus Karloff was able to show that the ratio exp=opt can be arbitrarily close to ff. However, this does not show the ....

....It is an interesting open question whether some fixed d (e.g. d = 3 ) suffices for all values of ffl. Another issue of importance is the distribution of points on the sphere. Graphs that are composed of the n = 2 d points represented as the vectors 1 p d f Sigma1g d were considered in [13, 1]. Corollary 15 in Section 4.2 shows that these so called hypercubes embeddings cannot be used to demonstrate an integrality ratio other than 1. This fact was known prior to this paper. For our construction the number of points will also be exponential in d, but they will be distributed at ....

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H. Karloff. "How good is the Goemans-Williamson MAX CUT algorithm?". SIAM J. Comput., Vol. 29, No. 1, pp.336--350, 1999.

....are given and the label S stands for SDP, e.g. S100 = 100 trials with algorithm SDP. 13 0.48 0.5 0.52 0.54 0.56 0.58 0.6 R1 R5 R10 R50 R100 R500 R1000 p=0.7,q=0.0,without local opt. R=Rand Figure 5: Random applied to a bipartite graph without local optimization. As already shown in [8], we saw that when SDP was applied to graphs of the class GBiRand , it always computed the optimum in terms of SDPValDual. The number of trials did not influence the result in this case. On the other hand, the results which were computed by Random were worse than the ones given by SDP. Therefore, ....

H. J. Karloff, How good is the Goemans-Williamson MaxCut algorithm?, Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 96), 427-434, 1996.

....15 For a graph with no negative edge weights, Goemans and Williamson [63] give the currently best approximation algorithm for this problem in 1994. They use the technique of semidefinite relaxation to significantly improve the approximation ratio of MAXCUT to 0. 878, which is provably tight [88]. For a graph with negative edge weights, Goemans and Williamson s algorithm guarantees to find a cut whose value minus the negative edge weights is no less than 0.878 times the maximum cut value minus the negative edge weights. Their result is a major breakthrough in designing approximation ....

....Their result is a major breakthrough in designing approximation algorithms in that for the first time they successfully adapt a nonlinear programming technique to design approximation algorithm. Much research, including the results described in this thesis, was prompted by their pioneering work [86, 55, 51, 43, 94, 95, 104, 78, 111, 6, 30, 88, 98]. There are also some results on approximation algorithms for MAXCUT on special graphs. It is known that MAXCUT has a polynomial time approximation scheme when the given graph is dense [49, 10] 2.2.2 COLORING COLORING [84] has applications in register allocation, time tabling, scheduling, ....

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In ACM [3], pages 427--434.

....ratio [4] 3 On Random Hyperplanes One way of proving the conjecture is by a rounding technique that converts any optimal SDP solution to a true cut of value at least 0:879 times the SDP value. The random hyperplane rounding technique can not serve for this purpose, as shown by Karloff [7]. We will give a simplified presentation of a slightly weaker result. Consider the 7 cycle. The value of the maximum cut is 6. There are seven cuts that achieve this value, where each such cut looses one particular edge (v i ; v i 1 ) We have seen that the optimal value for the SDP is also 6. A ....

H. Karloff. "How good is the Goemans-Williamson MAX CUT Algorithm". In Proc. 28th STOC, 427--434, 1996. 5

....in many di erent ways, including as an SDP. Schrijver [17] then showed that, for association schemes, a re nement of the theta function (with nonnegativity constraints added to one of the formulations) is equivalent to the LP bound of Delsarte. In the context of the maximum cut problem, Karlo [7] and Alon and Sudakov [1] have also exploited simpli cations of SDP for instances that arise from association schemes. In this note we will derive the following two results. SDP over an association scheme, or more generally, SDP where the input matrices commute, are equivalent to ordinary LP ....

....They give a randomized algorithm producing a cut whose expected value E[cut] satis es E[cut] z psd ; where = 2 min 0 1 cos 0:87856 whenever W 0.This implies that z mc z psd . Although it is not known whether zmc z psd can be arbitrarily close to , Karlo [7] has proved that, for graphs arising from the Johnson (association) scheme, E[cut] z psd can be arbitrarily close to . In [6] it was shown that the performance guarantee of can be improved whenever the maximum cut is known to be a large fraction of the total weight of the edges (see [6] for ....

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H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm. In Proc. of the 28th ACM Symp. on Theory Comput., pages 427-434, 1996.

....relaxation (3) for Max Cut, Goemans and Williamson proved the following. Theorem 24 ( 22] There exists a polynomial time algorithm that provides two numbers z L ; z sdp such that z L z mc z sdp and z L z sdp :87856 hold for all graphs G on nonnegative edge weights. Later Karloff [34] showed that the analysis of the rounding algorithm in [22] can not be improved. On the negative side, there is a recent result of Hastad, showing that it is difficult to approximate Max Cut tightly. Theorem 25 ( 23] Theorem 4.2) For any 0 it is NP hard to approximate (undirected) MaxCut ....

H. KARLOFF. How good is the Goemans-Williamson MAX-CUT algorithm? in: Proceedings 28th Symposium on the Theory of Computer Science, 1996, 427--434.

....CUT. The worst case value for OPT=SDP is thus somewhere between 0:87856 and 0:88446, and even though this gap is small, it would be very interesting to prove Delorme and Poljak s conjecture that the worst case is given by the 5cycle. This would, however, require a new technique. Indeed, Karloff [31] has shown that the analysis of the random hyperplane technique is tight, namely there exists a family of graphs for which the expected weight E[w(ffi(S) of the cut produced is arbitrarily close to ff SDP . Instead of comparing (13) and (11) term by term, Nesterov [52] recently proposed a ....

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm. In Proc. of the 28th ACM Symp. on Theory Comput., pages 427--434, 1996.

....and an appropriate (randomized) rounding technique. It is proved in [5] that the approximation guarantee of this algorithm is at least the minimum of the function h(t) t in (0; 1] where h(t) 1 arccos(1 Gamma 2t) This minimum is attained at t 0 = 0:844: and is roughly 0:878. Karloff [7] showed that this minimum is indeed the correct approximation guarantee of the algorithm, by constructing appropriate graphs. The authors of [5] also proved that their algorithm has a better approximation guarantee for graphs with large cuts. If A t 0 , with t 0 as above, and the maximum cut of ....

....1 Gamma j 2 : The rest of this short paper is organized as follows. In Section 2 we prove Theorem 1.1 and present examples showing that its statement is optimal, up to a constant factor. In Section 3 we prove Theorem 1.2 by constructing appropriate graphs. Our construction resembles the one in [7], but is more general and its analysis is somewhat simpler. The main advantage of the new construction is that unlike the one in [7] it is a Cayley graph of an abelian group and therefore its eigenvalues have a simple expression, and can be compared with each other without too much efforts. The ....

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H. Karloff, How good is the Goemans-Williamson MAX CUT algorithm?, Proc. of the 28 th ACM STOC, ACM Press (1996), 427--434.

....is to seek to optimize over the set E n M n . This was rst proposed in [19] Adding triangle inequalities to SDP1 is considered in e.g. 8, 9, 10] However, it is not the case that adding a certain subset of the triangle inequalities will improve every instance of max cut. Furthermore, Karlo [13] has shown that it is impossible to improve the performance guarantee of Goemans and Williamson simply by adding valid linear inequalities to SDP1. In this paper we present an SDP relaxation for MC which improves on the idea of adding triangle inequalities to SDP1. The feasible set of this tighter ....

H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm ? SIAM J. on Computing, 29(1):336-350, 1999.

....in many different ways, including as an SDP. Schrijver [17] then showed that, for association schemes, a refinement of the theta function (with nonnegativity constraints added to one of the formulations) is equivalent to the LP bound of Delsarte. In the context of the maximum cut problem, Karloff [7] and Alon and Sudakov [1] have also exploited simplifications of SDP for instances that arise from association schemes. In this note we will derive the following two results. ffl SDP over an association scheme, or more generally, SDP where the input matrices commute, are equivalent to ordinary ....

....They give a randomized algorithm producing a cut whose expected value E[cut] satisfies E[cut] z psd ff; where ff = 2 min 0 1 Gammacos 0:87856 whenever W 0.This implies that z mc ffz psd . Although it is not known whether zmc z psd can be arbitrarily close to ff, Karloff [7] has proved that, for graphs arising from the Johnson (association) scheme, E[cut] z psd can be arbitrarily close to ff. In [6] it was shown that the performance guarantee of ff can be improved whenever the maximum cut is known to be a large fraction of the total weight of the edges (see [6] ....

[Article contains additional citation context not shown here]

H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm. In Proc. of the 28th ACM Symp. on Theory Comput., pages 427--434, 1996.

....constraints to add to the SDP relaxation (2.17) i.e. one can try and add the so called triangle inequalities which are the standard inequalities used in branch and bound methods for MC. This has proven to be very successful in practice, e.g. 39] However, one cannot guarantee an improvement, [42]. But, in the space of matrices, it is also true that X 2 = xx T xx T = nX: Therefore we can use the following equivalent quadratic matrix model for MCQ. max Trace QX s.t. diag(X) e X 2 Gamma nX = 0; 3.32) where X is a symmetric matrix. Note that a common diagonalization of X ....

H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm? SIAM J. on Computing, 29(1):336--350, 1999.

No context found.

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, Philadelphia, Pennsylvania, pages 427--434, 1996.

No context found.

H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm ? SIAM J. on Computing, 29(1):336--350, 1999.

No context found.

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm. In Proc. of the 28th ACM Symp. on Theory Comput., pages 427--434, 1996.

No context found.

Karloff, H.: How Good is the Goemans-Williamson MAX CUT Algorithm?, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC96), pp. 427-434, 1996.

No context found.

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, Philadelphia, Pennsylvania, pages 427--434, 1996.

No context found.

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, Philadelphia, Pennsylvania, pages 427--434, 1996.

No context found.

H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? Manuscript, 1995.

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H. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, Philadelphia, Pennsylvania, pages 427--434, 1996.

No context found.

Karloff, H.: How Good is the Goemans-Williamson MAX CUT Algorithm?, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC96), pp. 427-434, 1996.

No context found.

H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm? SIAM J. on Computing, 29(1):336--350, 1999.

No context found.

H. KARLOFF. How good is the Goemans-Williamson MAX CUT algorithm? In Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996.

No context found.

H. J. Karloff. How good is the Goemans-Williamson MAX CUT algorithm? In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 427--434, 1996.

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