On the complexity of integer multiplication in branching programs with multiple tests and in read-once branching programs with limited nondeterminism

Branching programs (BPs) are a well-established computation and representation model for Boolean functions. Although exponential lower bounds for restricted BPs such as read-once branching programs (BP1s) have been known for a long time, the proof of lower bounds for important selected functions is sometimes difficult. Especially the complexity of fundamental functions such as integer multiplication in different BP models is interesting. In (Bolling and Woelfel, 2001), the first strongly exponential lower bound of Ω(2^n/4) has been proven for the complexity of integer multiplication in the deterministic BP1 model. Here, we consider two well-studied BP models which generalize BP1s by allowing a limited amount of nondeterminism and multiple variable tests, respectively. More precisely, we prove a lower bound of Ω(2ⁿ(7k)/) for the complexity of integer multiplication in the (V, k)-BP model. As a corollary, we obtain that integer multiplication cannot be represented in polynomial size by nondeterministic BP1s, if the number of nondeterministic nodes is bounded by log n - log log n - ω (1). Furthermore, we show that any (1, +k)-BP representing integer multiplication has a size of Ω(2[n/48(k+1)]). This is not polynomial for k = o(n/log n)