Showing papers by "Johan Håstad published in 2010"

An efficient parallel repetition theorem

Abstract: We present a general parallel-repetition theorem with an efficient reduction. As a corollary of this theorem we establish that parallel repetition reduces the soundness error at an exponential rate in any public-coin argument, and more generally, any argument where the verifier’s messages, but not necessarily its decision to accept or reject, can be efficiently simulated with noticeable probability.

On the List-Decodability of Random Linear Codes

Abstract: For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\epsilon > 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1-H_q(p)-\epsilon)n$ has the property that every Hamming ball of radius $pn$ has at most $O(1/\epsilon)$ codewords. This answers a basic open question concerning the list-decodability of linear codes, showing that a list size of $O(1/\epsilon)$ suffices to have rate within $\epsilon$ of the "capacity" $1-H_q(p)$. Our result matches up to constant factors the list-size achieved by general random codes, and gives an exponential improvement over the best previously known list-size bound of $q^{O(1/\epsilon)}$. The main technical ingredient in our proof is a strong upper bound on the probability that $\ell$ random vectors chosen from a Hamming ball centered at the origin have too many (more than $\Theta(\ell)$) vectors from their linear span also belong to the ball.

On the list-decodability of random linear codes

Abstract: The list-decodability of random linear codes is shown to be as good as that of general random codes. Specifically, for every fixed finite field Fq, p ∈ (0,1 - 1/q) and e >; 0, it is proved that with high probability a random linear code C in Fqn of rate (1-Hq(p)-e) can be list decoded from a fraction p of errors with lists of size at most O(1/e). This also answers a basic open question concerning the existence of highly list-decodable linear codes, showing that a list-size of O(1/e) suffices to have rate within e of the information-theoretically optimal rate of 1 - Hq(p). The best previously known list-size bound was qO(1/e) (except in the q = 2 case where a list-size bound of O(1/e) was known). The main technical ingredient in the proof is a strong upper bound on the probability that I random vectors chosen from a Hamming ball centered at the origin have too many (more than Ω(l)) vectors from their linear span also belong to the ball.

Approximating linear threshold predicates

Abstract: We study constraint satisfaction problems on the domain {-1, 1}, where the given constraints are homogeneous linear threshold predicates That is, predicates of the form sgn(w1x1 ++ wnxn) for some positive integer weights w1, , wn Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not The focus of this paper is to identify and study the approximation curve of a class of threshold predicates that allow for non-trivial approximation Arguably the simplest such predicate is the majority predicate sgn(x1 ++ xn), for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture Our techniques extend to a more general class of "majoritylike" predicates and we obtain parallel results for them In order to classify these predicates, we introduce the notion of Chow-robustness that might be of independent interest