Randomness versus Nondeterminism for Read-Once and Read-k Branching Programs

Abstract

Recent breakthroughs have lead to strong methods for proving lower bounds on the size of branching programs (BPs) with quite weak restrictions. Nevertheless, lower bounds for the randomized and nondeterministic variants of the established BP models still offer many challenges. Here, the knowledge on the randomized case is extended as follows:

(i) The so-far open problem of proving that randomization with arbitrary bounded error can be weaker than nondeterminism for read-once BPs is solved in the following strong sense: It is shown that the so-called “weighted sum function” requires strongly exponential size for randomized read-once BPs with error bounded by any constant smaller than 1/2, while both the function and its complement have polynomial size for nondeterministic read-once BPs.

(ii) For randomized read-k BPs, an exponential lower bound for a natural, graph-theoretical function that is easy to compute nondeterministically is presented. This is the first such bound for the boolean BP model. The function cl3,n deciding whether an n-vertex graph contains a triangle is obviously easy for nondeterministic read-once BPs while its complement is known to require strongly exponential size in this model. It is proved here that the function still requires size 2ω(k ^-22^-4k√n) for randomized read-k BPs with error at most 2^{-c2 2k} for some positive constant c.

Supported by DFG grant We 1066/9.