Improved Polynomial Identity Testing for Read-Once Formulas

An <em>arithmetic read-once formula</em> (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are { +,×} and such that every input variable labels at most one leaf. In this paper we study the problems of giving deterministic identity testing and reconstruction algorithms for ROFs. Our main result is an $n^{{\mathcal{O}}(k + \log n)}$ time deterministic algorithm for checking whether a black box holding the sum of <em>k</em> <em>n</em> -variate ROFs computes the zero polynomial. In other words, we provide a hitting set of size $n^{{\mathcal{O}}(k + \log n)}$ for the sum of <em>k</em> ROFs. This result greatly improves [27] where an $n^{{\mathcal{O}}(k^2 + \sqrt n)}$ algorithm was given for the problem.

Using our new results we obtain a deterministic reconstruction algorithms for read-once formulas that runs in time $n^{{\mathcal{O}}(\log n)}$.

In fact, our results also hold for the more general model of <em>preprocessed</em> read-once formulas that we define in this paper. In this model we are allowed to replace each variable <em>x</em> _<em>i</em> with a polynomial <em>T</em> _<em>i</em> (<em>x</em> _<em>i</em> ).

Our techniques are very close to the techniques in [27]. The main difference is that we obtain several tighter versions of the tools first used there. In particular we obtain a better version of the <em>hardness of representation</em> approach which was first used in [27]. This technique can be thought of as a very explicit way of transforming (mild) hardness of a very structured polynomial to an identity testing algorithm.