How Bad is the Freedom to Flood-It?

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

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Rémy Belmonte
  • The University of Electro-Communications, Chofu, Tokyo, Japan
Mehdi Khosravian Ghadikolaei
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR, LAMSADE, 75016 Paris, France
Masashi Kiyomi
  • Yokohama City University, Yokohama, Japan
Michael Lampis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR, LAMSADE, 75016 Paris, France
Yota Otachi
  • Kumamoto University, Kumamoto, Japan

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Rémy Belmonte, Mehdi Khosravian Ghadikolaei, Masashi Kiyomi, Michael Lampis, and Yota Otachi. How Bad is the Freedom to Flood-It?. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
  • flood-filling game
  • parameterized complexity


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