Bipartite perfect matching is in quasi-NC

Computer Science/Discrete Mathematics Seminar I
Topic:Bipartite perfect matching is in quasi-NC
Speaker:Stephen Fenner
Affiliation:University of South Carolina
Date:Monday, February 8
Time/Room:11:15am - 12:15pm/S-101
Video Link:https://video.ias.edu/csdm/2016/0208-Fenner

We show that the bipartite perfect matching problem is in $\textrm{quasi-}\textsf{NC}^2$. That is, it has uniform circuits of quasi-polynomial size and $O(\log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the Isolation Lemma of Mulmuley, Vazirani, & Vazirani, used to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem. Time permitting, we describe an $\textsf{RNC}^2$ algorithm to find a perfect matching in a bipartite graph using $O(\log^2 n)$ random bits. Joint work with Rohit Gurjar and Thomas Thierauf (Aalen University).