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 2c/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 Θ(√ cn). 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. Abstract: We consider a class of highly dynamic networks modelled on an urban subway system. We examine the problem of creating a map of such a subway in less than ideal conditions, where the local residents are not enthusiastic about the process and there is a limited ability to communicate amongst the mappers. More precisely, we study the problem of a team of asynchronous computational entities (the mapping agents) determining the location of black holes in a highly dynamic graph, whose edges are defined by the asynchronous movements of mobile entities (the subway carriers). We present and …
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