R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science, pages 280--285, Los Angeles, October 1987. IEEE.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....discernible, or else most users are su#ciently indi#erent to the clustering that we can make good recommendations without isolating them explicitly. These considerations reflect a sense in which our goals here are quite di#erent from what one encounters in problems with planted structure (e.g. [3, 4]) while we do have an underlying generative model, it is possible to perform well even in cases where the underlying structure is provably unrecoverable. Since the Weighted Model includes the Uniform Model, a special case of Theorem 1.1 yields a near optimal recommendation algorithm for the ....

R. Boppana, "Eigenvalues and graph bisection: An average-case analysis," Proc. IEEE Symp. on Foundations of Computer Science, 1987.

.... instances with k = #( # n log n) and instances with # = #(n 1 2 # ) Furthermore, under the requirements of Ben Dor et al. the running time of our algorithm is O(n log n) For the case of two equal sized clusters, our algorithm handles almost the same range of # as the algorithm of Boppana [2], but our algorithm is faster, and is also more general since it handles as many as m = #( # n log n) clusters. We note that Table 1 cites results as given in the papers, even though in several cases better results can be obtained by improving the analysis or by making small modifications to the ....

....k =##3 1 # n log n # 2 3 n 2 3 ) For the case of equal sized clusters, our algorithm has a wider range of k when k # Ben Dor et al. 1] n) ##23 O(n This study ##u 1 # # n log n) ## n 1 2 # ) # O(mn (a) General case. m # Dyer Frieze [6] 2 ## n 1 4 log 1 4 n) O(n Boppana [2] 2 ## # pn 1 2 # log n) n Jerrum Sorkin [8] 2 ## n 1 6 # ) O(n Jules [9] 2 ##23 O(n Condon Karp [5] O(1) n 1 2 # ) O(n Carson Impagliazzo [3] 2 #( # pn 1 2 log n) O(n Feige Kilian [7] 2 ## # pn 1 2 # log n) n This study O( # n log n) # ## mn 1 2 log n) O( 1)n ....

R. B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proc. 28th Symposium on Foundation of Computer Science (FOCS 87), pages 280--285, 1987.

.... with random graphs drawn from the so called planted bisection model for which it has been theoretically shown [7] that there is little difference between the simple evolutionary algorithm (1 1) ES and the more sophisticated EAs such as the Metropolis algorithm, and a problem specific heuristic [4]. Often the EAs are used in situations where the information on quantitative parameters of the input problem (except for its dimensionality) is accessible only through evaluations of objective function in the sample points. This raises the issue of evaluation and comparison of different algorithms ....

R.B.Boppana (1987) Eigenvalues and Graph Bisection: an Average Case Analysis. In Proc. of the 28th IEEE Symposium on Foundations of Computer Science, 280-285.

....graph spectra. Early theoretical work connecting graph spectra and partitioning is due to Barnes, Donath and Hoffman [1] 6] 7] More recent eigenvector and eigenvalue methods have dealt with both module placement (Frankle and Karp [11] and Tsay and Kuh [41] and graph min cut bisection (Boppana [3] and Blanks [2] In general, these previous works formulate the partitioning problem as the assignment or placement of nodes The reader will note that our discussion omits several popular approaches, including simulated annealing [22] 38] genetic algorithms [24] and relaxation based ....

R. B. Boppana. Eigenvalues and graph bisection: An average case analysis. In IEEE Symp. on Foundations of Computer Science, pages 280--285, 1987.

....are quite often the basis for load balancing in parallel computing. Hendrickson and Leland [18] provide a good overview on static graph partitioning algorithms. They include greedy algorithms like the famous one by Kernighan and Lin [23] and extensions of it [5] so called spectral methods [2, 4, 1] and other hybrid approaches. Several of these algorithms have been implemented and integrated into tools that can be used as off line pre processors to partitioning a central computing problem. Among these tools are Chaco [19] METIS [22] and JOSTLE [40] Recent research also tackles the problem ....

R. B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proceedings of the twenty eighth Annual Symposium on Foundations of Computer Science, pages 280285, Los Angeles, USA, 1987.

....3.2 Let G be a weighted graph of order n. If n is even, then If n is odd, then 4n The bounds of Corollary 3.2 can be further improved by introducing a correction function. A function c is called a correction function if c 1. The following bound was proved by Boppana [Bo]. Proposition 3.3 Let G be a weighted graph of even order n. Then diag(c) f, f# (20) and the minimum is taken over all nonzero functions f with f 1. V (G) be a set of cardinality with e(S, S) bw(G) and let g be its signed characteristic function, i.e. ....

R. B. Boppana, Eigenvalues and graph bisection: An average case analysis, 28th Annual Symp. Found. Comp. Sci., IEEE, 1987, pp. 280--285.

....is also known as balanced bi partitioning, or bi section problem. Because of the NP hardness of the bi partitioning problem [8] heuristics have a crucial role for the solution of large size partitioning problems in acceptable computing times. The problem has been studied extensively in the past [2, 3, 5]. In particular, 9] presents a detailed empirical study of Simulated Annealing (SA) following the application proposed in [11, 12] Simple Genetic Algorithms (GA) are proposed in [14, 15] while a state of the art GA with preliminary preprocessing is 1991 Mathematics Subject Classification. ....

R. B. Boppana, Eigenvalues and graph bisection: an average-case analysis, Proc. 28th Symp. Foundations of Computer Science, 1987, pp. 280--285.

....bisection of a graph, and Fiedler [11] 12] who explored the properties of # 2 and its associated eigenvector for the Laplacian. There has been much subsequent work, including Barnes s partitioning algorithm [5] Boppana s work that included a stronger lower bound on the minimum bisection size [6], work by Rendl, Wolkowicz, and others using optimization approaches [24, 10] and the particular bisection and graph partitioning methods considered in this paper [18] 23] 25] Since our work first appeared [17] Spielman and Teng [27] have extended the latter methods to include recursion. It ....

R. Boppana, Eigenvalues and graph bisection: An average-case analysis, in 28th Annual Symposium on Foundations of Computer Science, Los Angeles, October 1987, IEEE, pp. 280-- 285.

....defined by the chosen partition, and with probability p if it does not. If q is sufficiently smaller than p, then the bisection defined by the chosen partition is, almost surely, a minimum bisection. Several heuristics for this and similar planted bisection models were studied in [BCLS87, Bop87, DF89, JS98, CK99, FK01a] Semi random input models. Although the input distributions employed by random models are quite natural, there is usually no claim that these models represent instances that occur in real life applications. Furthermore, a heuristic that relies excessively on statistical ....

....for the maximum independent set problem. Evidence for optimality. An average case algorithm does not have an apriori guarantee on its performance, and it is therefore valuable to certify that the solution it produced on the particular instance at hand is indeed optimal. The algorithm of Boppana [Bop87] for the minimum bisection problem has such a certification property (see also [FK01a] His algorithm outputs a bisection together with a lower bound (that is obtained by a relaxation) on the minimum cost of a bisection. If the cost of the output bisection is equal to the lower bound then it is ....

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R. B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science, pages 280--285, October 1987. 87

....combined with understanding of the information provided by these parameters can constitute a very powerful tool, capable of solving algorithmic problems where all other methods failed. This is especially true for randomly generated graphs, several successful examples of spectral techniques are [6], 2] 3] A survey [1] discusses several applications of spectral techniques to graph algorithms. In order to show that bad graphs have an exponentially small probability in G(n; p) we will prove a new large deviation result for eigenvalues of random symmetric matrices. This result, bounding ....

R. Boppana, Eigenvalues and graph bisection: An average case analysis, Proc. 28 th IEEE FOCS, IEEE (1987), 280-285.

....himself, using Algorithm B. The spectral properties of a graph encode some detailed structural information on it. The ability to compute the eigenvectors and eigenvalues of a graph in polynomial time provides a powerful algorithmic tool, which has already found several applications (see, e.g. [7], 2] 22] The spectral approach, and the techniques developed here, may well have additional algorithmic applications in the future too. ....

R. Boppana, Eigenvalues and graph bisection: An average case analysis, Proc. of the 28 th IEEE FOCS, IEEE (1987), 280-285.

....and so on. We shall thus consider only the problem of partitioning the graph into two equal pieces. There are several ways to solve approximately this graph bisection problem, and two popular algorithms are: spectral bisection, and geometric methods such as inertial bisection. The spectral method [1,2,7,9] uses properties of the task graph itself, such as the Laplacian matrix of the graph. In contrast, geometric methods [9] make a statistical hypothesis, assuming that each task is associated with a definite position in some geometric space, and ignore the detailed structure of 3 the task ....

R. B. Boppana, Eigenvalues and Graph Bisection: an Average Case Analysis, in 28th Annual Symp. Found. Comp. Sci, 1987.

....Dyer and Frieze [6] analyze an algorithm for (not necessarily regular) graphs with Omega Gamma n 2 ) edges and b (1 Gamma ffl)m=2 for a fixed ffl 0. The Dyer Frieze algorithm is based on comparison of vertex degrees; it finds the minimum bisection in polynomial expected time. Boppana [5] presents a graph bisection algorithm based on eigenvector methods. He shows that if m is Omega Gamma n log n) and b (m Gamma 5 p mn log n) 2, then his algorithm finds the minimum bisection with probability 1 Gamma O(1=n) Thus, Boppana s analysis applies to a larger class of graphs than the ....

R. B. Boppana. "Eigenvalues and graph bisection: an average-case analysis," in Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, 1987, 280--285.

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R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science, pages 280--285, Los Angeles, October 1987. IEEE.

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R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science, pages 280--285, Los Angeles, October 1987. IEEE. 29

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R. Boppana. Eigenvalues and graph bisection: An average case analysis. In Proc. on Foundations of Computer Science, pages 280--285, 1987. 7

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Ravi B. Boppana. Eigenvalues and graph bisection: An average case analysis. In Proc. 28th Annual FOCS, 1987.

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R. Boppana. Eigenvalues and graph bisection: An average case analysis. In 28th Annual Symposium on Foundations of Computer Science, October 12--14, 1987.

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Ravi Boppana. Eigenvalues and graph bisection: An average-case analysis. In FOCS, 1987.

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R. B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proc. 28th Symposium on Foundation of Computer Science (FOCS 87), pages 280--285, 1987.

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R. B. Boppana. Eigenvalues and graph bisection: an average case analysis. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages. 280--285, 1987.

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R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Ashok K. Chandra, editor, Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pages 280-285, 1987.

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R.B. Boppana, "Eigenvalues and Graph Bisection: An AverageCase Analysis," Proc. 28th Ann. IEEE Symp. Foundations of Computer Science, pp. 280-285, 1987.

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R.B. Boppana. Eigenvalues and Graph Bisection: An Average-Case Analysis. In Proc. of 28th Annual FOCS, pp. 280-285, 1987.

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R.B. BOPPANA. Eigenvalues and graph bisection: An average case analysis. In Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pages 280--285, Los Angeles, California, 1987. IEEE.

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