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 “lowdegree 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 SelfReduction for LowDegree 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 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
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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 “dimensionreduction” trick.
Let’s 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
≤
δ
=
1
n
2
.
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 Spring '09
 MadhuSudan
 Computational complexity theory, Degree of a polynomial, zp

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