Parameterized Complexity of Flood-Filling Games on Trees

Abstract

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. As for many colored graph problems, Flood-filling games have relevant interpretations in bioinformatics. The standard versions of Flood-It and Free-Flood-It are played on n ×m grids. In this paper we analyze the complexity of these games when played on trees. We prove that Flood-It remains NP-hard on trees whose leaves are at distance at most d = 2 from the pivot, and that Flood-It is in FPT when parameterized by the number of colors c in such trees (for any constant d). We also show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, keeping complexity features; this implies that Flood-It on trees inherits several complexity results already proved for RSCS, such as some interesting FPT and W[1]-hard cases. We introduce a new variant of Flood-It, called Multi-Flood-It, where each move of the game is played on various pivots. We also present a general framework for reducibility from Flood-It to Free-Flood-It, by defining a special graph operator ψ such that Flood-It played on a graph class $\mathcal{F}$ is reducible to Free-Flood-It played on the image of $\mathcal{F}$ under ψ. An interesting particular case occurs when $\mathcal{F}$ is closed under ψ. Some NP-hard cases for Free-Flood-It on trees can be derived using this approach. We conclude by showing some results on parameterized complexity for Free-Flood-It played on pc-trees (phylogenetic colored trees). We prove that some results valid for Flood-It on pc-trees can be inherited by Free-Flood-It on pc-trees, using another type of reducibility framework.