Hardness Ampli ation Within NP

Texts in Computational Complexity: Ampli cation of Hardness

Pseudorandom Generators without the XOR Lemma

Impagliazzo andWigderson IW have recently shown that if there exists a decision problem solvable in time O n and having circuit complexity n for all but nitely many n then P BPP This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators The construction of Impagliazzo andWigderson goes through ...

On the Worst-Case Approximability of Sparse PCA

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1. a simple and efficient algorithm that achieves an n-approximation; 2. NP-hardness of approximation to within (1 − ε), for some small constant ε > 0; 3. SSE-hardness of approximation t...

CS 880: Advanced Complexity Theory Final Project: Hardness Amplification within NP

We survey the current status of the problem of Hardness amplification within NP, reviewing known results and evaluating various approaches for improving on them. We also prove that under some stronger assumptions, we can amplify the hardness to (12 − 1/2).

On the Approximability of Sparse PCA

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1. a simple and efficient algorithm that achieves an n−1/3-approximation; 2. NP-hardness of approximation to within (1− ε), for some small constant ε > 0; 3. SSE-hardness of approximatio...