## Abstract

We study the complexity of the popular one player combinatorial game known as Flood-It. In this game the player is given an *n*×*n* board of tiles where each tile is allocated one of *c* colours. The goal is to make the colours of all tiles equal via the shortest possible sequence of flooding operations. In the standard version, a flooding operation consists of the player choosing a colour *k*, which then changes the colour of all the tiles in the monochromatic region connected to the top left tile to *k*. After this operation has been performed, neighbouring regions which are already of the chosen colour *k* will then also become connected, thereby extending the monochromatic region of the board. We show that finding the minimum number of flooding operations is **NP**-hard for *c*≥3 and that this even holds when the player can perform flooding operations from any position on the board. However, we show that this ‘free’ variant is in **P** for *c*=2. We also prove that for an unbounded number of colours, Flood-It remains **NP**-hard for boards of height at least 3, but is in **P** for boards of height 2. Next we show how a (*c*−1) approximation and a randomised 2*c*/3 approximation algorithm can be derived, and that no polynomial time constant factor, independent of *c*, approximation algorithm exists unless **P**=**NP**. We then investigate how many moves are required for the ‘most demanding’ *n*×*n* boards (those requiring the most moves) and show that the number grows as fast as \(\Theta(\sqrt{c}\, n)\). Finally, we consider boards where the colours of the tiles are chosen at random and show that for *c*≥2, the number of moves required to flood the whole board is Ω(*n*) with high probability.

## Keywords

NP-completeness Flood-filling Combinatorial games Percolation## Preview

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