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

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

Google Summer of Code 2020: D-Complete and Mobile Posets

-
In this post, I thought I'd overview the work of  my first paper , which has just recently been accepted to SIDMA.  In order theory, a linear extension of a partially ordered set P is a totally ordered set L which respects the original ordering P. Many famous problems can be converted into counting the number of linear extensions. For example, the number of standard Young tableaux of a given shape is equivalent to the number of linear extensions of a poset with the same shape. Unfortunately, it is #P-complete in general to count linear extensions (even for posets of height 2!). However, there are famous classes of posets like Young diagrams (or more generally, d-complete posets), which have efficient computation. In our paper, we presented a new class of posets called mobiles that have determinant formulas for counting linear extensions and generalize d-complete posets and ribbon posets. Since this past summer converted me into a shut-in, I decided to spend some of that extra time
Read more

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