We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an N-vertex graph to within a factor of N ɛ is NP-hard. 1

We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called ffl-biased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are ffl-biased can be used to construct "almost" k-wise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k- wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using ffl-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...

We show that any randomized algorithm that runs in space S and time T and uses poly(S) random bits can be simulated using only O(S) random bits in space S and time T poly(S). A deterministic simulation in space S follows. Of independent interest is our main technical tool: a procedure which extracts randomness from a defective random source using a small additional number of truly random bits. 1

This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight non-approximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.

We show that modified versions of the linear congruential generator and the shift register generator are provably good for amplifying the correctness of a probabilistic algorithm. More precisely, if r random bits are needed for a BPP algorithm to be correct with probability at least 2=3, then O(r + k 2 ) bits are needed to improve this probability to 1 \Gamma 2 \Gammak . We also present a different pseudo-random generator that is optimal, up to a constant factor, in this regard: it uses only O(r + k) bits to improve the probability to 1 \Gamma 2 \Gammak . This generator is based on random walks on expanders. Our results do not depend on any unproven assumptions. Next we show that our modified versions of the shift register and linear congruential generators can be used to sample from distributions using, in the limit, the information-theoretic lower bound on random bits. 1. Introduction Randomness plays a vital role in almost all areas of computer science, both in theory and in...

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A new technique, based on the pseudo-random properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For any fixed (small) rate, and for sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF (2)) as well. Although these concatenated codes lie below Zyablov bound, they are still superior to previously-known explicit constructions in the zero-rate neighborhood.

We show how to simulate BPP and approximation algorithms in polynomial time using the output from a ffi-source. A ffi-source is a weak random source that is asked only once for R bits, and must output an R-bit string according to some distribution that places probability no more than 2 \GammaffiR on any particular string. We also give an application to the unapproximability of Max Clique.

this paper we do two things. First, we survey extractors and dispersers: what they are, how they can be designed, and some of their applications. The work described in the survey is due to a long list of research papers by various authors##most notably by David Zuckerman. Then, we present a new tool for constructing explicit extractors and give two new constructions that greatly improve upon previous results. The new tool we devise, a merger," is a function that accepts d strings, one of which is uniformly distributed and outputs a single string that is guaranteed to be uniformly distributed. We show how to build good explicit mergers, and how mergers can be used to build better extractors. Using this, we present two new constructions. The first construction succeeds in extracting all of the randomness from any somewhat random source. This improves upon previous extractors that extract only some of the randomness from somewhat random sources with enough" randomness. The amount of truly random bits used by this extractor, however, is not optimal. The second extractor we build extracts only some of the randomness and works only for sources with enough randomness, but uses a nearoptimal amount of truly random bits. Extractors and dispersers have many applications in removing randomness" in various settings and in making randomized constructions explicit. We survey some of these applications and note whenever our new constructions yield better results, e.g., plugging our new extractors into a previous construction we achieve the first explicit N-superconcentrators of linear size and polyloglog(N) depth. ] 1999 Academic Press CONTENTS 1.

.<F3.827e+05> We consider a finite random walk on a weighted graph<F3.539e+05><F3.827e+05> G; we show that the fraction of time spent in a set of vertices<F3.539e+05> A<F3.827e+05> converges to the stationary probability<F3.539e+05><F3.827e+05><F3.539e+05><F3.827e+05> #(A) with error probability exponentially small in the length of the random walk and the square of the size of the deviation from<F3.539e+05><F3.827e+05><F3.539e+05><F3.827e+05> #(A). The exponential bound is in terms of the expansion of<F3.539e+05> G<F3.827e+05> and improves previous results of [D. Aldous,<F3.405e+05> Probab. Engrg. Inform.<F3.827e+05> Sci., 1 (1987), pp. 33--46], [L. Lovasz and M. Simonovits,<F3.405e+05> Random Structures<F3.827e+05> Algorithms, 4 (1993), pp. 359--412], [M. Ajtai, J. Komlos, and E. Szemeredi,<F3.405e+05> Deterministic simulation of<F3.827e+05> logspace, in Proc. 19th ACM Symp. on Theory of Computing, 1987]. We show that taking the sample average from one trajectory gives a more e#cien...