Derandomization of BPP Using Hitting Set Generators

-
In 1999, Goldreich, Vadhan, and Wigderson wrote a beautiful and simple paper that the derandomization of RP through hitting set generators also derandomizes BPP.

In the following write up, I survey basic results on RP and BPP, as well as the results of this paper. If you are too lazy to read, feel free to watch a recorded lecture I made on the topic while quarantined at my grandfather's house!

If you are wondering why I look like I am in a witness protection program in the recording, it is cause my DIY whiteboard fails with any internal lighting.

Comments

Post a Comment

Popular posts from this blog

The Catalan Grenade is Accepting Papers!

-
As an inaugural post to this blog, I will announce that I am perpetually accepting papers! Here at the Catalan Grenade, I'll be going through papers on cool results, old and new, in Combinatorics and Algebra. Currently, I am a recent graduate from UMass Amherst (Fall 2018), and am waiting to hear back from grad schools. This seems like a fun way to pass the time.
Read more

Extension Complexity Part II - Classifying Nonnegative Rank by Randomized Communication

-
A classical result from Yannakakis states that $$\text{xc}(P) = \text{rank}_+(S),$$ where \(P\) is a polytope with slack matrix \(S\). Yannakakis further showed that with nondeterministic communication complexity, one can give a lower bound for the nonnegative rank. This lower bound is sufficient to prove that some polytopes, like the correlation polytope, have exponential extension complexity. However, it fails for many other examples.  In a 2013 paper, "Extended formulations, nonnegative factorizations, and randomized communication protocols," by Faenza, Fiorini, Grappe, and Tiwary, they show that a model of randomized communication complexity can not only lower bound nonnegative rank, but exactly characterize it.  In my communication complexity reading group, I gave a talk presenting this paper and its results. The proofs are extremely elegant and simple, making for a very enjoyable read.  For those interested, here is a link to my lecture notes! 
Read more

Number-on-the-Forehead Communication Complexity

-
 Since the Winter 2020 semester at McGill, I've helped run a Communication Complexity reading group attended by undergraduate and graduate students. Recently, I became particularly excited by the Number-on-the-Forehead model, and gave a talk on it in the reading group.  The idea is quite simple. For 2 player communication, imagine that Alice's input is on top of Bob's forehead, making it easy for Alice to see but impossible for Bob to see. Likewise, Bob's input is on Alice's forehead.  This generalizes to k-player problems where k people stand in a circle, each with a card on their forehead indicating some information that is viewable to everyone but the forehead owner.  If this still doesn't make sense, think of this scene from The Office.  Here are the  lecture notes  I wrote for the talk. I cover basic definitions, Behrend's bound and it's application to the Exactly-N problem, and Ramsey Theoretic lower bounds on cylinders.  As I liked this topic so
Read more