MCSP is Hard for Read-Once Nondeterministic Branching Programs

Abstract

We show that every read-once nondeterministic branching program computing the Minimum Circuit Size Problem on inputs of length N has size $\varOmega (N^{\log \log (N)})$. This is the first superpolynomial lower bound on the size of computing ${\textsc {MCSP} }$. This lower bound is tight for the version of ${\textsc {MCSP} }$ restricted to a linear circuit size parameter.

To show this result we adapt a conditional lower bound of Ilango [10] on the deterministic Turing Machine time complexity of computing ${\textsc {MCSP} }^{*}$, the generalization of ${\textsc {MCSP} }$ to partial functions. In contrast, our lower bound is unconditional and holds even for the total ${\textsc {MCSP} }$ function.

En route, we get two results that may be of independent interest:

• The size of the minimal computing ${\textsc {MCSP} }$ equals, up to a constant factor, the size of the minimal computing ${\textsc {MCSP} }^{*}$;

• The size of any computing $(2n \times 2n)$-Bipartite Independent Set is $\varOmega (n!)$.