Positive semidefinite programs are an important subclass of semidefinite programs in which all matrices involved in the specification of the problem are positive semidefinite and all scalars involved are non-negative. We present a parallel algorithm, which given an instance of a positive semidefinite program of size N and an approximation factor ε > 0, runs in (parallel) time poly( 1 ε ) · polylog(N), using poly(N) processors, and outputs a value which is within multiplicative factor of (1 + ε) to the optimal. Our result generalizes analogous result of Luby and Nisan [10] for positive linear programs and our algorithm is inspired by the algorithm of [10].
We show a parallel repetition theorem for the entangled value ω * (G) of any two-player one-round game G where the questions (x, y) ∈ X × Y to Alice and Bob are drawn from a product distribution on X × Y. We show that for the k-fold product G k of the game G (which represents the game G played in parallel k times independently)
A strong direct product theorem for a problem in a given model of computation states that, in order to compute k instances of the problem, if we provide resource which is less than k times the resource required for computing one instance of the problem with constant success probability, then the probability of correctly computing all the k instances together, is exponentially small in k. In this paper, we consider the model of two-party bounded-round public-coin randomized communication complexity. For a relation f ⊆ X ×Y ×Z (X , Y, Z are finite sets), let R A fundamental question in complexity theory is how much resource is needed to solve k independent instances of a problem compared to the resource required to solve one instance. More specifically, suppose for solving one instance of a problem with probability of correctness p, we require c units of some resource in a given model of computation. A natural way to solve k independent instances of the same problem is to solve them independently, which needs k · c units of resource and the overall success probability is p k . A strong direct product theorem for this problem would state that any algorithm, which solves k independent instances of this problem with o(k · c) units of the resource, can only compute all the k instances correctly with probability at most p −Ω(k) .In this work, we are concerned with the model of communication complexity which was introduced by Yao [Yao79]. In this model there are different parties who wish to compute a joint relation of their inputs. They do local computation, use public/private coins, and communicate between them to achieve this task. The resource that is counted is the number of bits communicated. The text by Kushilevitz and Nisan [KN96] is an excellent reference for this model. Direct product questions and the weaker direct sum questions have been extensively investigated in different sub-models of communication complexity. A direct sum theorem states that in order to compute k independent instances of a problem, if we provide resources less than k times the resource required to compute one instance of the problem with the constant success probability p < 1, then the success probability for computing all the k instances correctly is at most a constant q < 1. Some examples of known direct product theorems are: Parnafes, Raz and Wigderson's [PRW97] theorem for forests of communication protocols; Shaltiel's [Sha04] theorem for the discrepancy bound (which is a lower bound on the distributional communication complexity) under the uniform distribution; extended to arbitrary distributions by Lee, Shraibman andŠpalek [LSv08]; extended to the multiparty case by Viola and Wigderson [VW08]; extended to the generalized discrepancy bound by Sherstov [She11]; Jain, Klauck and Nayak's [JKN08] theorem for subdistribution bound; Klauck,Špalek, de Wolf's [KŠdW04] theorem for the quantum communication complexity of the set disjointness problem; Klauck's [Kla10] theorem for the public-coin communication complexity of the set-disjointness...
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