Bootstrapping variables in algebraic circuits

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One needs only to consider size-s degree-s circuits that depend on the firstlog○c svariables (where c is a constant and composes a logarithm with itself c times). Thus, the hitting-set generator (hsg) manifests a bootstrapping behavior—a partial hsg against very few variables can be efficiently grown to a complete hsg. A Boolean analog, or a pseudorandom generator property of this type, is unheard of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w.r.t. variables. This is repeated c times in an efficient way. Pushing the envelope further we show that (i) a quadratic-time blackbox PIT for 6,913-variate degree-s size-s polynomials will lead to a “near”-complete derandomization of PIT and (ii) a blackbox PIT for n-variate degree-s size-s circuits insnδtime, forδ<1/2, will lead to a near-complete derandomization of PIT (in contrast,sntime is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable,O(s1.49)-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of Kabanets and Impagliazzo [Kabanets V, Impagliazzo R (2003)Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing STOC ’03].

Parallel algorithms for matroid intersection and matroid parity

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).