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More on average case vs approximation complexity. (English) Zbl 1242.68109
Summary: We consider the problem to determine the maximal number of satisfiable 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 algorithms and in cryptography. Namely, we show that our conjectures imply the following facts:
\(\circ\)
Feige’s hypothesis about the hardness of refuting a random 3CNF is true, which in turn implies inapproximability within a constant for several combinatorial problems, for which no NP-hardness of approximation is known.
\(\circ\)
It is hard to approximate the nearest codeword within factor \(n ^{1-\epsilon }\).
\(\circ\)
It is hard to estimate the rigidity of a matrix. More exactly, it is hard to distinguish between matrices of low rigidity and random ones.
\(\circ\)
There exists a secure public-key (probabilistic) cryptosystem, based on the intractability of decoding of random binary codes.

MSC:
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
68W25 Approximation algorithms
03F20 Complexity of proofs
68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)
94A60 Cryptography
Software:
McEliece
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