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# lect16 - 6.841 Advanced Complexity Theory April 9 2007...

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6.841 Advanced Complexity Theory April 9, 2007 Lecture 16 Lecturer: Madhu Sudan Scribe: Amanda Redlich, Shubhangi Saraf 1 Overview In today’s lecture we will cover the following Permanent: worst case average case Permanent IP towards IP PSPACE 2 The Permanent We begin be recalling the definition of the permanent of a matrix. It is a function of n 2 integers. Permanent ( ( x ij ) n i,j =1 ) = X Π S n n Y i =1 x i Π( i ) We observe that the permanent of an n × n matrix is a degree n polynomial in n 2 variables. It is a “low-degree multivariate polynomial,” and this turns out to be a very interesting feature of the permanent, as we’ll see in the rest of the lecture. 3 Random Self-Reduction for Low-Degree Poly- nomials This problem was first studied by Beaver–Feigenbaum and Lipton. The basic question is as follows: We are given a polynomial f ( x 1 , . . . , x m ) of degree d over some finite field F = Z p (say). Given an algorithm that computes f on random instances, can we compute f ( a 1 , . . . , a m ) for any worst-case a 1 , a 2 , . . . , a m ? There are two ways of understanding what it means to have an algorithm that computes f on random instances. When each input x i ; for 1 i m is chosen independently and uniformly from Z p , either for most instances, say for ( 1 - 1 n 2 ) fraction, the algorithm outputs the right answer, and for 1 n 2 fraction it gives the wrong answer, or 16-1

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the algorithm runs in expected polynomial time for the given distribution, and always outputs the right answer. The first notion is weaker than the second, since we know that we can easily convert an algorithm of the second kind to the first kind. We’ll assume the weaker notion when we talk about an algorithm that computes f on random instances. 4 Worst Case to Average Case Reduction for the Low Degree Polynomials Let f be an m -variate polynomial, with degree d . Then f has ( d + m m ) = ( d + m d ) coefficients. Since this number is exponentially large, it is not tractable to write down all the terms of the polynomial explicitly. To get around this problem, we employ the “dimension-reduction” trick. Let’s say that we have a poly-time algorithm A that evaluates f correctly on most inputs. Assume that for x Z m p , f ( x ) = A ( x ), unless x B , where | B | p m δ = 1 n 2 .
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lect16 - 6.841 Advanced Complexity Theory April 9 2007...

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