Lecture 3: Hardness Amplification
نویسندگان
چکیده
Remark 1. One may wonder why Impagliazzo’s hard core lemma (see the previous lecture for the lemma statement) is not enough to get a 12 −2 −δn-hard function starting with a 2−δn-hard function. The difficulty lies in the fact that the hard core lemma only tells about the existence of a hard core set such that on that set f is 1 2 − 2 −δn -hard. We do not know how to sample from that hard core set efficiently.
منابع مشابه
Cs 880: Advanced Complexity Theory Lecture 13: Average-case Hardness 1 Worst-case vs. Average-case Complexity
In this lecture and the next two lectures we study hardness amplification, in which the goal is to take a mildly average-case hard function from some class and construct another function in that class that is very average-case hard. Today we prove a lemma that roughly states that every average-case hard function has a set of inputs that encapsulates the hardness of that function in a certain se...
متن کاملCS 880 : Advanced Complexity Theory 2 / 29 / 2008 Lecture 15 : Hardness Amplification within NP
In the last lecture, we introduced the general idea of boosting the hardness of a function by taking k independent copies of the function and aggregating them using another function h. We obtained the following result:
متن کاملCMSC 858F: Algorithmic Lower Bounds: Fun with Hardness Proofs Fall 2014 Quadratic Hardness and the 3-SUM Problem
In the previous lecture, we looked at the APSP problem and some of the other closely related problems. We studied the cubic hardness of these problems. In this lecture, we will go about doing something similar, but in the domain of quadratic hardness. With regard to this, we will choose 3-SUM problem as the representative problem. We will look at some related problems, that can be reduced to th...
متن کاملCMSC 858 F : Algorithmic Lower Bounds Fall 2014 Puzzles and Reductions from 3 - Partition
In this lecture, we first examine several classes of NP-hardness and polynomial time algorithms which arise from differences in how integers are encoded in problem input. We then look at the 3-partition problem, which is very useful for proving the strongest notion of NP-hardness. Finally, we use reduction from 3-partition to prove NP-hardness for a handful of problems, including a set of 4 pac...
متن کاملA Theorist ’ s Toolkit ( CMU 18 - 859 T , Fall 2013 ) Lecture 24 : Hardness Assumptions
This lecture is about hardness and computational problems that seem hard. Almost all of the theory of hardness is based on assumptions. We make assumptions about some problems, then we do reductions from one problem to another. Then, we want to make the minimal number of assumptions necessary to show computational hardness. In fact, all work on computational complexity and hardness is essential...
متن کامل