TY - GEN

T1 - Explicit dimension reduction and its applications

AU - Karnin, Zohar S.

AU - Rabani, Yuval

AU - Shpilka, Amir

PY - 2011

Y1 - 2011

N2 - We construct a small set of explicit linear transformations mapping ℝn to ℝt, where t = O (log(γ-1) ε-2), such that the L2 norm of any vector in Rn is distorted by at most 1 ± ε in at least a fraction of 1 - γ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is sub-optimal compared with probabilistic arguments, we nevertheless are able to apply our construction to a number of problems. In particular, we use it to construct an ε-sample (or pseudo-random generator) for linear threshold functions on Sn-1, for ε = o(1). We also use it to construct an ε-sample for spherical digons in Sn-1, for ε = o(1). This construction leads to an efficient oblivious derandomization of the Goemans-Williamson MAX CUT algorithm and similar approximation algorithms (i.e., we construct a small set of hyperplanes, such that for any instance we can choose one of them to generate a good solution). Our technique for constructing ε-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on k-wise independent sample spaces.

AB - We construct a small set of explicit linear transformations mapping ℝn to ℝt, where t = O (log(γ-1) ε-2), such that the L2 norm of any vector in Rn is distorted by at most 1 ± ε in at least a fraction of 1 - γ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is sub-optimal compared with probabilistic arguments, we nevertheless are able to apply our construction to a number of problems. In particular, we use it to construct an ε-sample (or pseudo-random generator) for linear threshold functions on Sn-1, for ε = o(1). We also use it to construct an ε-sample for spherical digons in Sn-1, for ε = o(1). This construction leads to an efficient oblivious derandomization of the Goemans-Williamson MAX CUT algorithm and similar approximation algorithms (i.e., we construct a small set of hyperplanes, such that for any instance we can choose one of them to generate a good solution). Our technique for constructing ε-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on k-wise independent sample spaces.

UR - http://www.scopus.com/inward/record.url?scp=80051970609&partnerID=8YFLogxK

U2 - 10.1109/CCC.2011.20

DO - 10.1109/CCC.2011.20

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AN - SCOPUS:80051970609

SN - 9780769544113

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 262

EP - 272

BT - Proceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011

T2 - 26th Annual IEEE Conference on Computational Complexity, CCC 2011

Y2 - 8 June 2011 through 10 June 2011

ER -