Abstract
Read- k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ABP). In this work, we give an exponential lower bound of exp ( n/k O ( k ) ) on the width of any read- k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial-size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2 Õ( n 1−1/2 k −1 ) and needs white box access only to know the order in which the variables appear in the ABP.
Highlights
Algebraic complexity studies the complexity of syntactically computing polynomials using arithmetic operations
For the class of k-pass algebraic branching program (ABP), we provide a black-box polynomial identity testing (PIT) algorithm that runs in quasipolynomial time
We show that by discarding more variables, but not too many, we get a structure that we call a “k-regularly interleaving sequence”. This is a technical notion which is presented in full details in Section 5, but the main point is that this definition allows us to argue that the obtained read-k oblivious ABP has a evaluation dimension and it can be simulated by a not-too-large read-once oblivious ABP (ROABP)
Summary
Algebraic complexity studies the complexity of syntactically computing polynomials using arithmetic operations. Two of the most important problems in algebraic complexity are (i) proving exponential lower bounds for arithmetic circuits (i.e., proving that any circuit computing some explicit polynomial f must be of exponential size), and (ii) giving an efficient deterministic algorithm for the polynomial identity testing (PIT) problem. It is not hard to show that ROABPs are strictly stronger than read-once arithmetic formulas Another motivation to study this model is that it is the algebraic analog of a boolean read-once branching program, which arises in the context of pseudorandomness for small-space computation [51]. We consider the natural step, which are read-k oblivious algebraic branching programs This model generalizes and extends both the models of ROABPs, of read-k arithmetic formulas and of sum of ROABPs. We are able to prove exponential lower bounds and to give subexponential-time PIT algorithms for this model.
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