Abstract
Fixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic (flooded) graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight.
Highlights
Flood-It is a popular puzzle, originally released as a computer game in 2006 by LabPixies
The player has the right to change the color of all vertices contained in the same monochromatic component as the pivot to a different color of her choosing
Doing this judiciously gradually increases the size of the pivot’s monochromatic component, until the whole graph is flooded with one color
Summary
Flood-It is a popular puzzle, originally released as a computer game in 2006 by LabPixies (see [1]). Our first result is to show that Free-Flood-It is W[2]-hard parameterized by the number of moves in an optimal solution.
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