A Note on Amortized Branching Program Complexity

TL;DR

This paper demonstrates that for any function, there exists a highly efficient branching program computing many copies of it, challenging previous conjectures and providing new insights into branching program complexity and lower bounds.

Contribution

It introduces a construction of compact branching programs that compute multiple copies of any function, disproving a conjecture and impacting complexity theory techniques.

Findings

Almost all functions require exponential size branching programs.

Existence of linear-size branching programs for many copies of any function.

Disproof of a conjecture about non-uniform catalytic computation.

Abstract

In this paper, we show that while almost all functions require exponential size branching programs to compute, for all functions $f$ there is a branching program computing a doubly exponential number of copies of $f$ which has linear size per copy of $f$. This result disproves a conjecture about non-uniform catalytic computation, rules out a certain type of bottleneck argument for proving non-monotone space lower bounds, and can be thought of as a constructive analogue of Razborov's result that submodular complexity measures have maximum value $O(n)$.