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- W3021570316 abstract "We revisit the problem of hardness amplification in $mathcal{NP}$, as recently studied by O'Donnell [J. Comput. System Sci., 69 (2004), pp. 68-94]. We prove that if $mathcal{NP}$ has a balanced function f such that any circuit of size $s(n)$ fails to compute f on a $1/poly(n)$ fraction of inputs, then $mathcal{NP}$ has a function $f'$ such that any circuit of size $s'(n)=s(sqrt{n})^{Omega(1)}$ fails to compute $f'$ on a $1/2 - 1/s'(n)$ fraction of inputs. In particular, begin{enumerate} item if $s(n)=n^{omega(1)}$, we amplify to hardness $1/2-1/n^{omega(1)}$; item if $s(n)=2^{n^{Omega(1)}}$, we amplify to hardness $1/2-1/2^{n^{Omega(1)}}$; item if $s(n)=2^{Omega(n)}$, we amplify to hardness $1/2-1/2^{Omega(sqrt{n})}$. end{enumerate} Our results improve those of of O'Donnell, which amplify to $1/2-1/sqrt{n}$. O'Donnell also proved that no construction of a certain general form could amplify beyond $1/2-1/n$. We bypass this barrier by using both derandomization and nondeterminism in the construction of $f'$. We also prove impossibility results demonstrating that both our use of nondeterminism and the hypothesis that f is balanced are necessary for black-box hardness amplification procedures (such as ours)." @default.
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- W3021570316 date "2006-01-01" @default.
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- W3021570316 title "Using Nondeterminism to Amplify Hardness" @default.
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- W3021570316 doi "https://doi.org/10.1137/s0097539705447281" @default.
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