Rohit Gurjar

Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

We give an

Bipartite perfect matching is in quasi-NC

n^{O(\log n)}

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

-time (

Linear matroid intersection is in quasi-NC

n

Deterministic identity testing for sum of read-once oblivious arithmetic branching programs

is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious arithmetic branching programs (ROABPs). The best time complexity known for blackbox polynomial identity testing (PIT) for this class was

Identity testing for constant-width, and commutative, read-once oblivious ABPs

n^{O(\log^2 n)}

Identity Testing for Constant-Width, and Any-Order, Read-Once Oblivious Arithmetic Branching Programs

due to Forbes, Saptharishi, and Shpilka [Proceedings of the 2014 ACM Symposium on Theory of Computing, 2014, pp. 867--875]. Moreover, their result holds only when the individual degree is small, while we do not need any such assumption. With this, we match the time complexity for the unknown-order ROABP with the known-order ROABP (due to Forbes and Shpilka [Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013, pp. 243--252]) and also with the depth-3 set-multilinear circuits (due to Agrawal, Saha, and Saxena [Proceedings of the 2013 ACM Symposium on Theory of Computing, 2013, pp. 321--330]). Our proof is simpler and involves a new technique called basis isolation. The depth-3 ...

Planarizing Gadgets for Perfect Matching Do Not Exist

We show that the bipartite perfect matching problem is in quasi- NC2. That is, it has uniform circuits of quasi-polynomial size nO(logn), and O(log2 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 famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

Guest Column: Parallel Algorithms for Perfect Matching

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial-time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial-time complexity