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Unformatted text preview: 6.841 Advanced Complexity Theory April 9, 2007 Lecture 16 Lecturer: Madhu Sudan Scribe: Amanda Redlich, Shubhangi Saraf 1 Overview In todays 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 S n n Y i =1 x i ( i ) We observe that the permanent of an n n matrix is a degree n polynomial in n 2 variables. It is a lowdegree multivariate polynomial, and this turns out to be a very interesting feature of the permanent, as well see in the rest of the lecture. 3 Random SelfReduction for LowDegree Poly nomials This problem was first studied by BeaverFeigenbaum 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 worstcase 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 161 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. Well 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 mvariate 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 dimensionreduction trick. Lets say that we have a polytime 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 = 1 n 2 ....
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.
 Spring '09
 MadhuSudan

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