The complexity of flood-filling games on graphs

TL;DR

This paper investigates the computational complexity of flood-filling games on graphs, providing polynomial algorithms for certain cases and proving NP-hardness for others, revealing the nuanced difficulty depending on graph structure and coloring.

Contribution

It introduces a polynomial time algorithm for linking vertex pairs and analyzes the complexity of flooding paths and boards, establishing NP-hardness for specific configurations.

Findings

Polynomial algorithm for linking vertex pairs in graphs

Efficient computation of flooding moves for paths and fixed-width boards

NP-hardness of flooding for 3x n boards with four or more colors

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. Although computing the minimum number of moves required to flood an arbitrary graph is known to be NP-hard, we demonstrate a polynomial time algorithm to compute the minimum number of moves required to link each pair of vertices. We apply this result to compute in polynomial time the minimum number of moves required to flood a path, and an additive approximation to this quantity for an arbitrary k x n board, coloured with a bounded number of colours, for any fixed k. On the other hand, we show that, for k>=3, determining the minimum number of moves required to flood a k x n board coloured with at least four colours remains NP-hard.