S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28(5):1641--1663, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... applied, with modi cations, to several other problems in combinatorial optimization (e.g. 3, 4, 8, 9, 17, 22, 27, 30] Almost any randomized approximation algorithm based on the hyperplane technique applied to a semide nite relaxation can be derandomized, as was shown by Mahajan and Ramesh [21]. In the meantime, researchers in mathematical programming have shown that the interior point methods that extend polynomial time solvability from linear programming to semide nite programming also extend polynomial time solvability to the class of symmetric cones. This includes the second order ....

....randomized ( approximation algorithm for instances of Max 2 Lin Mod 3 that have inequations only, where = 7 12 3 ( 1=4) 0:836008: This implies a randomized 0:836008 approximation algorithm for Max 3 Cut. Our randomized algorithm seems to t the framework of Mahajan and Ramesh [21] and we therefore expect that it can be derandomized using their approach. Previously, Andersson, Engebretsen, and H astad [3] considered a more general variant of the problem Max 2 Lin Mod p (for two variable equations mod p) and shown that they could obtain an ( p) approximation ....

S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semide nite programming. SIAM Journal on Computing, 28:1641-1663, 1999.

....Max Cut uses randomization. Is this an intrinsic property of the algorithm, or is it possible to #nd a deterministic approximation algorithm with the same performance Goemans and Williamson [40] proposed a simple derandomization scheme, but it was soon discovered to be faulty. Mahajan and Ramesh [70] later presented a derandomization scheme for approximation algorithms based on semide#nite programming, thus providing derandomization of Goemans and Williamson s Max Cut algorithm as well as other algorithms. This result may be interesting from a purely theoretical point of view, but in practice ....

Sanjeev Mahajan and Hariharan Ramesh. Derandomizing approximation algorithms based on semide#nite programming. SIAM Journal of Computing, 28(5):1641#1663, 1999.

....is deterministic so that it is guaranteed to produce a coloring using O(jSj ff k Gamma2 ) colors, if G[S] is (k Gamma 2) colorable. Our algorithm, however, is randomized. There are two ways of overcoming this difficulty. The first is to derandomize it using the technique of Mahajan and Ramesh [MR99]. Alternatively, we can simply repeat the whole algorithm a sufficient number of times so that the error probability is small enough. 6 Concluding remarks We obtained several improved coloring algorithms. It would be interesting to obtain further improvements. In particular, it would be ....

S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28:1641-- 1663, 1999.

....between MIN ENERGY and MAX CUT problems can be also used to design an approximation algorithm for the MIN ENERGY problem. Recently, a randomized approximation algorithm with a high performance guarantee = 0:87856 for the MAX CUT formulation (9) has been proposed [19] and later de randomized [45], which can straightforwardly be exploited for approximating the MIN ENERGY problem. Namely, for bipolar Hopfield nets it can be observed that MIN ENERGY can be approximated in a polynomial time within absolute error less than 0:243W [70] where W is the network weight (13) For W = O(s 2 ) e.g. ....

S. Mahajan and H. Ramesh, Derandomizing approximation algorithms based on semidefinite programming, SIAM Journal on Computing 28 (5) (1999) 1641--1663.

....and thus E[jI 0 j] n 1 2 N(c) n 1 2 1 c 1 c 3 1 p 2 e c 2 =2 = n 3 3 1 m 2 3 1 = n 3 3 m 2 3 ; as promised. 2 Note that even though our algorithm is randomized, it can be easily derandomized using the technique of Mahajan and Ramesh [20]. We will nish this section with the following immediate consequence of the last theorem. Corollary 4 If 1=2 and H is a 3 uniform hypergraph on n vertices with an independent set of size at least n, then there is a polynomial time algorithm which nds in H an independent set of size 786 ....

S. Mahajan and H. Ramesh, Derandomizing approximation algorithms based on semidenite programming, SIAM J. Computing 28 (1999), 1641-1663. 13

.... applied, with modifications, to several other problems in combinatorial optimization (e.g. 3, 4, 8, 9, 17, 22, 27, 30] Almost any randomized approximation algorithm based on the hyperplane technique applied to a semidefinite relaxation can be derandomized, as was shown by Mahajan and Ramesh [21]. In the meantime, researchers in mathematical programming have shown that the interior point methods that extend polynomial time solvability from linear programming to semidefinite programming also extend polynomial time solvability to the class of symmetric cones. This includes the second order ....

.... #) approximation algorithm for instances of Max 2 Lin Mod 3 that have inequations only, where # = 7 12 3 4# 2 arccos 2 ( 1 4) 0.836008. This implies a randomized 0.836008 approximation algorithm for Max 3 Cut. Our randomized algorithm seems to fit the framework of Mahajan and Ramesh [21] and we therefore expect that it can be derandomized using their approach. Previously, Andersson, Engebretsen, and Hastad [3] considered a more general variant of the problem Max 2 Lin Mod p (for two variable equations mod p) and shown that they could obtain an ( 1 p #(p) approximation ....

S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28:1641--1663, 1999.

....solvable. Furthermore, a polynomial time approximate algorithm that solves the MIN ENERGY problem to within absolute error of less than 0:243W in binary Hop eld nets of weight W has been introduced in [121] based on a high performance approximate algorithm for the MAX CUT problem [32] [83]. For W = O(s 2 ) e.g. for symmetric networks with s binary neurons and constant weights, this results matches the lower bound s 2 ) 11] which cannot be guaranteed by any approximate polynomial time MIN ENERGY algorithm for every 0, unless P = NP . In addition, the MIN ENERGY problem ....

S. Mahajan and H. Ramesh, Derandomizing approximation algorithms based on semidenite programming, SIAM Journal on Computing 28 (5) (1999) 1641-1663.

....Theorem 5.4. If Conjecture 5.3 is valid, computing an almost optimal solution to (SDP ) modifying it according to f 2 , and using Algorithm Random Hyperplane to construct a feasible schedule yields a randomized approximation algorithm with expected performance guarantee 1:3388. It is shown in [Mahajan and Ramesh 1999] that Algorithm Random Hyperplane can be derandomized. We get a deterministic version of our approximation algorithm if we make use of the derandomized version of Algorithm Random Hyperplane. We can also apply Algorithm Randomized Rounding to turn a feasible solution a = a(v) to (SDP ) into a ....

Mahajan, S. and Ramesh, H. 1999. Derandomizing approximation algorithms based on semidenite programming. SIAM Journal on Computing 28, 1641 - 1663.

....issue is polynomial time solvable. Further, a polynomial time approximate algorithm that solves the MIN ENERGY problem within absolute error less than 0:243W in binary Hop eld nets of weight W have been introduced [116] which is based on highperformance approximate algorithm for MAX CUT problem [31, 81]. For W = O(s 2 ) e.g. for symmetric networks with s binary neurons and constant weights, this results matches the lower bound s 2 ) which cannot be guaranteed by any approximate polynomial time MIN ENERGY algorithm for every 0 (unless P = NP ) 11] In addition, MIN ENERGY can be ....

S. Mahajan and H. Ramesh, Derandomizing approximation algorithms based on semidenite programming, SIAM Journal on Computing 28 (5) (1999) 1641{ 1663. 27

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28(5):1641--1663, 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28:1641--1663, 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28:1641-- 1663, 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28:1641--1663, 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semide nite programming. SIAM Journal on Computing, 28:1641{ 1663, 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semide nite programming. SIAM Journal on Computing, 28:1641-1663, 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28(5):1641--1663 (electronic), 1999.

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S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semide nite programming. SIAM Journal on Computing, 28:1641-1663, 1999.

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