Partial Minimum Branching Program Size Problem is ETH-hard

TL;DR

This paper proves that under the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem is superpolynomially hard, extending to various subclasses and establishing new lower bounds and hardness results.

Contribution

It establishes ETH-hardness for the partial minimization of branching programs and related subclasses, and shows NP-hardness of BP size compression verification.

Findings

ETH implies superpolynomial lower bounds for MBPSP*

Unconditional superpolynomial lower bounds for read-once nondeterministic BPs

NP-hardness of BP size compression decision problem

Abstract

We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem (MBPSP*) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs. Combining these results with the recent unconditional lower bounds for MCSP [Glinskih, Riazanov'22], we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1-NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems. Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.