How Bad is the Freedom to Flood-It?

Belmonte, Rémy; Khosravian Ghadikolaei, Mehdi; Kiyomi, Masashi; Lampis, Michael; Otachi, Yota

Belmonte, Rémy; Khosravian Ghadikolaei, Mehdi; Kiyomi, Masashi; Lampis, Michael; Otachi, Yota (2019), How Bad is the Freedom to Flood-It?, Journal of Graph Algorithms and Applications, 23, 2, p. 111-134. 10.7155/jgaa.00486

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Type

Article accepté pour publication ou publié

Date

2019

Journal name

Journal of Graph Algorithms and Applications

Volume

Number

Publisher

Brown University

Pages

111-134

Abstract (EN)

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.

Subjects / Keywords

flood-filling game; parameterized complexity