Two types of branching programs with bounded repetition that cannot efficiently compute monotone 3-CNFs

TL;DR

This paper establishes exponential lower bounds for certain classes of non-deterministic branching programs computing monotone 3-CNFs, advancing understanding of computational complexity for these logical formulas.

Contribution

It introduces two new classes of branching programs with bounded repetition and proves exponential lower bounds for their ability to compute specific monotone 3-CNFs.

Findings

Exponential lower bounds for non-deterministic branching programs on certain monotone 3-CNFs.

Lower bounds hold for programs with repetition up to logarithmic in input size.

Progress in understanding the complexity of higher-repetition branching programs for 2-CNF related formulas.

Abstract

It is known that there are classes of 2-CNFs requiring exponential size non-deterministic read-once branching programs to compute them. However, to the best of our knowledge, there are no superpolynomial lower bounds for branching programs of a higher repetition computing a class of 2-CNFs. This work is an attempt to make a progress in this direction. We consider a class of monotone 3-CNFs that are almost 2-CNFs in the sense that in each clause there is a literal occurring in this clause only. We prove exponential lower bounds for two classes of non-deterministic branching programs. The first class significantly generalizes monotone read-$k$-times {\sc nbp}s and the second class generalizes oblivious read $k$ times branching programs. The lower bounds remain exponential for $k \leq \log n/a$ where $a$ is a sufficiently large constant.