A. V. Goldberg, S. A. Plotkin, D. Shmoys, and ' E. Tardos. Interior-Point Methods in Parallel Computation. SIAM J. Comput., 21(1):149--150, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the problem due to Lov asz. This is particularly interesting because presently no linear programming formulation of the stable set and clique problems for perfect graphs with polynomially bounded number of facets is known. Linear programming interior point methods have been used by Goldberg et al. [GPST91] to derive sublinear time parallel algorithms for the bounded weight assignment problem. We show that maximum stable sets for perfect graphs can be computed in randomized sublinear parallel time. Furthermore, based on the work of Lov asz and Schrijver [LS91] we argue that in a branch and bound ....

A. V. Goldberg, S. A. Plotkin, D. Shmoys, and ' E. Tardos. Interior-Point Methods in Parallel Computation. SIAM J. Comput., 21(1):149--150, 1991.

.... [LN93] Hochbaum s original algorithm can be parallelized by employing Luby and Nisan s algorithm; the resulting algorithm would obtain an approximation ratio comparable to ours and have an incomparable running time (growing linearly with 1=ffl, but not with r) Previously, Goldberg et al. [GPST92] gave a parallel primal dual algorithm to find (exactly) maximum weight bipartite matchings. Their algorithm appears to be the first parallel algorithm to use primal dual techniques, but it requires polynomial time. 1.2 Problem Definitions Let G = V; E 2 V ) be a given hypergraph with ....

A. Goldberg, S. Plotkin, D. Shmoys, ' E. Tardos. Interior point methods in parallel computation. SIAM Journal on Computing, 21(1):140--150, 1992.

.... the fastest parallel maximum matching algorithm known for general graphs runs in O(n log n) time [13] Restricting the input to bipartite graphs allows us to find a maximum matching in O(n 2=3 log 3 n) time [14] which was improved to O( p m polylog(n) for sparse graphs (m 2 o(n 4=3 ) [15]. The only known NC maximum matching algorithms are for classes of graphs more restricted than the bipartite graphs, such as regular bipartite graphs [16] or planar bipartite graphs, where a perfect matching can be found [17] Curiously, fast deterministic algorithms for finding matchings in ....

A.V. Goldberg, S.K. Plotkin, D.B. Shmoys and ' E. Tardos, Interior point methods in parallel computation. in Proc. 30th Annual Symposium on Foundations of Computer Science, 1989, 350--355.

.... One application for such an implementation would be in the initial matching phase for a parallel algorithm for the assignment problem, such as the one described in [BMPT91] We would also like to note that parallel bipartite matching has received considerable attention from the theory community [Gro92, GPST92, GPV88, GT88a, SM89], but the algorithms described do not seem suitable for implementation. The remaining of the paper is divided in two parts: in the first (section 2) the sequential experiments are described, including implementations, test data and methodology, and results. In the second part (section 3) the ....

A. V. Goldberg, S. A. Plotkin, D. B. Shmoys, and E. Tardos. Interior point methods in parallel computation. SIAM J. Comput., February 1992.

....to Lov asz. This example is particularly interesting because presently no linear programming formulation of the stable set and clique problems for perfect graphs with polynomially bounded number of inequalities is known. Linear programming interior point methods have been used by Goldberg et al. [26] to derive sublinear time parallel algorithms for the bounded weight assignment problem. We show that maximum stable sets for perfect graphs can be computed in randomized sublinear parallel time. Furthermore, based on the work of Lov asz and Schrijver [42] we argue that in a branch and bound ....

A. V. Goldberg, S. A. Plotkin, D. Shmoys, and E. Tardos, Interior-Point Methods in Parallel Computation, SIAM J. Comput., 21 (1991), pp. 140--150.

.... Furthermore, similar to interior point methods for linear programming, the number of iterations used by this algorithm is proportional to the square root of the number of inequality constraints; for Lovasz number of graphs this turns out to be O ( p jV j) Following Goldberg et al. in [5], O (f(n) is synonymous with O(f(n) log k (n) for some constant k. This follows from the general results in the recent work of Nesterov and Nemirovsky [17] Once we have an oracle which computes (G) and therefore (G) in the case of perfect graphs, we may use a randomized technique based ....

....we may use a randomized technique based on the isolating lemma of Mulmuley, Vazirani and Vazirani [16] to construct the maximum clique and the maximum independent set in parallel. Our approach may be viewed somewhat loosely as a generalization of works of Goldberg, Plotkin, Shmoys and Tardos [5] and of Mulmuley, Vazirani and Vazirani [16] Goldberg et al. use an interior point linear programming technique to compute the maximum matching in bipartite graphs. The particular variant used there is based on the work by Ye [20] The authors achieve a deterministic O ( p jEj) parallel ....

A. Goldberg, S. Plotkin, D. Shmoys, and E. Tardos, Interior-Point Methods in Parallel Computation (Extended Abstract), Manuscript, April 1989.

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