The complexity of Free-Flood-It on 2xn boards

TL;DR

This paper analyzes the computational complexity of Free-Flood-It on 2xn boards, showing fixed parameter tractability with a bounded number of colours and NP-hardness when colours are unbounded.

Contribution

It proves that the problem is fixed parameter tractable for 2xn boards with a bounded number of colours and NP-hard otherwise, resolving an open question.

Findings

FPT algorithm for bounded colours on 2xn boards

NP-hardness for unbounded colours on 2xn boards

Resolution of an open complexity question

Abstract

We consider the complexity of problems related to the combinatorial game Free-Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. Our main result is that computing the length of an optimal sequence is fixed parameter tractable (with the number of colours present as a parameter) when restricted to rectangular 2xn boards. We also show that, when the number of colours is unbounded, the problem remains NP-hard on such boards. This resolves a question of Clifford, Jalsenius, Montanaro and Sach (2010).