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How Bad is the Freedom to Flood-It?

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Bibliographic Details
Title: How Bad is the Freedom to Flood-It?
Authors: Belmonte, Rémy, Khosravian Ghadikolaei, Mehdi, Kiyomi, Masashi, Lampis, Michael, Otachi, Yota
Contributors: Rémy Belmonte and Mehdi Khosravian Ghadikolaei and Masashi Kiyomi and Michael Lampis and Yota Otachi
Source: Journal of Graph Algorithms and Applications. 23:111-134
Publication Status: Preprint
Publisher Information: Journal of Graph Algorithms and Applications, 2019.
Publication Year: 2019
Subject Terms: FOS: Computer and information sciences, Programmation, Computer Science - Data Structures and Algorithms, Data Structures and Algorithms (cs.DS), 0102 computer and information sciences, ddc:004, flood-filling game, 01 natural sciences, logiciels, organisation des données, parameterized complexity
Description: ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ and ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ 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 ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ 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 ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ become intractable for ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$. We also show that some tractable cases for ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ are still tractable for ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ 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 ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ is always at most twice that of ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$, and this is tight.
Document Type: Article
Conference object
File Description: application/pdf
ISSN: 1526-1719
DOI: 10.7155/jgaa.00486
DOI: 10.48550/arxiv.1804.08236
DOI: 10.4230/lipics.fun.2018.5
Access URL: http://arxiv.org/abs/1804.08236
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.5
Rights: CC BY
arXiv Non-Exclusive Distribution
Accession Number: edsair.doi.dedup.....3007ca2f157d8b6ffad3b44dd378537e
Database: OpenAIRE
Description
Abstract:${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ and ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ 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 ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ 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 ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ become intractable for ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$. We also show that some tractable cases for ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ are still tractable for ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$ 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 ${\rm F{\small IXED-}F{\small LOOD-}I{\small T}}$ is always at most twice that of ${\rm F{\small REE-}F{\small LOOD-}I{\small T}}$, and this is tight.
ISSN:15261719
DOI:10.7155/jgaa.00486