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Summary: More on average case vs approximation complexity
Michael Alekhnovich
April 13, 2003
Abstract
We consider the problem to determine the maximal number of satis able equations
in a linear system chosen at random. We make several plausible conjectures about the
average case hardness of this problem for some natural distributions on the instances,
and relate them to several interesting questions in the theory of approximation algo-
rithms and in cryptography. Namely we show that our conjectures imply the following
facts:
Feige's hypothesis about the hardness of refuting a random 3CNF is true, which in
turn implies inapproximability within a constant for several combinatorial prob-
lems, for which no NP-hardness of approximation is known.
It is hard to approximate the NEAREST CODEWORD within factor n 1 .
It is hard to estimate the rigidity of a matrix. More exactly, it is hard to distin-
guish between matrices of low rigidity and random ones.
There exists a secure public-key (probabilistic) cryptosystem, based on the in-
tractability of decoding of random binary codes.
Our conjectures are strong in that they assume cryptographic hardness: no polyno-
mial algorithm can solve the problem on any non-negligible fraction of inputs. Never-
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