# Minimal Controllability of Conjunctive Boolean Networks is NP-Complete

## TL;DR

This paper investigates the minimal controllability problem for conjunctive Boolean networks, providing a controllability criterion, an efficient testing algorithm, and proving the problem's NP-hardness.

## Contribution

It offers a necessary and sufficient controllability condition, an $O(n^2)$ algorithm for testing, and proves NP-hardness of minimal controllability in CBNs.

## Key findings

- Controllability characterized by a specific condition.
- Efficient $O(n^2)$ algorithm for controllability testing.
- Minimal controllability problem is NP-hard.

## Abstract

Given a conjunctive Boolean network (CBN) with $n$ state-variables, we consider the problem of finding a minimal set of state-variables to directly affect with an input so that the resulting conjunctive Boolean control network (CBCN) is controllable. We give a necessary and sufficient condition for controllability of a CBCN; an $O(n^2)$-time algorithm for testing controllability; and prove that nonetheless the minimal controllability problem for CBNs is NP-hard.