4,961 research outputs found
The complexity of flood-filling games on graphs
We consider the complexity of problems related to the combinatorial game
Free-Flood-It, in which players aim to make a coloured graph monochromatic with
the minimum possible number of flooding operations. Although computing the
minimum number of moves required to flood an arbitrary graph is known to be
NP-hard, we demonstrate a polynomial time algorithm to compute the minimum
number of moves required to link each pair of vertices. We apply this result to
compute in polynomial time the minimum number of moves required to flood a
path, and an additive approximation to this quantity for an arbitrary k x n
board, coloured with a bounded number of colours, for any fixed k. On the other
hand, we show that, for k>=3, determining the minimum number of moves required
to flood a k x n board coloured with at least four colours remains NP-hard.Comment: More typos corrected
Extremal properties of flood-filling games
The problem of determining the number of "flooding operations" required to
make a given coloured graph monochromatic in the one-player combinatorial game
Flood-It has been studied extensively from an algorithmic point of view, but
basic questions about the maximum number of moves that might be required in the
worst case remain unanswered. We begin a systematic investigation of such
questions, with the goal of determining, for a given graph, the maximum number
of moves that may be required, taken over all possible colourings. We give
several upper and lower bounds on this quantity for arbitrary graphs and show
that all of the bounds are tight for trees; we also investigate how much the
upper bounds can be improved if we restrict our attention to graphs with higher
edge-density.Comment: Final version, accepted to DMTC
The complexity of Free-Flood-It on 2xn boards
We consider the complexity of problems related to the combinatorial game
Free-Flood-It, in which players aim to make a coloured graph monochromatic with
the minimum possible number of flooding operations. Our main result is that
computing the length of an optimal sequence is fixed parameter tractable (with
the number of colours present as a parameter) when restricted to rectangular
2xn boards. We also show that, when the number of colours is unbounded, the
problem remains NP-hard on such boards. This resolves a question of Clifford,
Jalsenius, Montanaro and Sach (2010)
How Bad is the Freedom to Flood-It?
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
Spanning trees and the complexity of flood-filling games
We consider problems related to the combinatorial game (Free-) Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of
flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time
How Bad is the Freedom to Flood-It?
International audienceFixed-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
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