### Using Nondeterminism to Amplify Hardness

Alexander Healy, Salil Vadhan, Emanuele Viola
2006 SIAM journal on computing (Print)
We revisit the problem of hardness amplification in N P, as recently studied by O'Donnell (STOC '02). We prove that if N P 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 N P has a function f such that any circuit of size s (n) = s( √ n) Ω(1) fails to compute f on a 1/2 − 1/s (n) fraction of inputs. In particular, 1. If s(n) = n ω(1) , we amplify to hardness 1/2 − 1/n ω(1) . 2. If s(n) = 2 n Ω(1) , we amplify to hardness
more » ... 2−1/2 n Ω(1) . 3. If s(n) = 2 Ω(n) , we amplify to hardness 1/2−1/2 Ω( √ n) . These improve the results of O'Donnell, which only amplified to 1/2 − 1/ √ 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).