Using Nondeterminism to Amplify Hardness
Alexander Healy, Salil Vadhan, Emanuele Viola · SIAM Journal on Computing · 2006
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).