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

Introducing Adaptive Flood Risk Management in England, New Zealand, and the Netherlands: The Impact of Administrative Traditions

Climate change adaptation creates significant challenges for decision makers in the flood risk-management policy domain. Given the complex characteristics of climate change, adaptive approaches(which can be adjusted as circumstances evolve) are deemed necessary to deal with a range of uncertainties around flood hazard and its impacts and associated risks. The question whether implementing adaptive approaches is successful highly depends upon how the administrative tradition of a country enable or hinder applying a more adaptive approach. In this article, we discern how the administrative tradition in the Netherlands, England, and New Zealand impact upon the introduction of adaptive flood risk management approaches. Using the concept of administrative traditions, we aim to explain the similarities and/or differences in how adaptive strategies are shaped and implemented in the three different state flood management regimes and furthermore, which aspects related to administrative traditions are enablers or barriers to innovation in these processe

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