Making Hard Problems Harder

We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function on a smaller number of bits which has greater hardness when measured in terms of input length. A hardness extractor takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function defined on a smaller number of bits which has close to maximum hardness. We prove several positive and negative results about these objects. First, we observe that hardness-based pseudo-random generators can be used to extract deterministic hardness from non-deterministic hardness. We derive several consequences of this observation. Among other results, we show that if E/O(n) has exponential non-determi...

• Type: Conference
• Year: 2006
• Where: COCO
• Authors: Joshua Buresh-Oppenheim, Rahul Santhanam
• Keywords: Algorithms, COCO 2006, Deterministic Hardness, Hardness, Hardness Condensing