2.6 Open Questions 47 We can identify all vertices with finite evacuation distance by starting with the safe vertices. Using the fire propagation preprocessing algorithm discussed in Section 2.3 to compute the ignition time of each vertex, we can identify all vertices with ignition time∞. These are all the vertices with an evacuation distance 0. A person starting on a neighbour of such a vertex can evacuate within a single round, except when it’s vertex was already burning at round 0. Thus, a vertex that did not ignite at round 0, and has a neighbours with evacuation distance 0, has evacuation distance (at most) 1. By the same argument, vertices that are still alive in roundk, and have a neighbour with evacuation distancek, have evacuation distance (at most)k+1. By these arguments, a single Breadth-First-Search from the safe vertices suffices to determine the evacuation distance of all vertices inG.
A firefighter might be especially interested in those vertices that do not have an evacuation path.
With saving as many people as possible as a priority, a firefighter would want to maximize the number of vertices that have evacuation paths. To do that he could protect some vertices to create new vertices to evacuate to, which gives rise to the following problem:
k-Evacuation Spot Problem Input:An instance of the r,e-model.
Goal: Find k vertices, such that protecting these vertices maximizes the number of vertices with evacuation distance <∞.
2.6.3 Budgeted Firefighting
A direct extension of Hartnell’s Firefighter as described in Section 2.2 is at least as hard as the original problem. However, the resistance and energy values of the vertices allows to define firefighting actions in a more nuanced way than just protecting a vertex entirely. Generally, two types of actions reduce the spread of the fire: On the one hand, increasing the resistance of vertices not yet on fire can cause them to ignite later or not at all, On the other hand, decreasing the energy of vertices, which already did or eventually will ignite, reduces their impact on adjacent vertices, causing them to ignite later or not at all.
So instead of protecting a vertex each round, the firefighter can increase the resistance or reduce the energy of one or more vertices, with the sum of those changes limited by some budgetb. The goal can also vary between minimizing the number of burned vertices, the number of rounds until the fire burns out, or even the number of vertices affected by the firefighters or the fire. The last option would be motivated by the fact that firefighting efforts themselves can cause damages, so we might want to apply them efficiently.
When reducing the objective to protecting a set of target vertices, we arrive at more general dynamic versions of the village protection problems discussed in Section 2.4, where increasing the resistance of the protective sets takes time. This will result in protective sets closer to the village cells to be better, as the distance to the fire allows for more time to apply the resistance increases. In addition, it might be necessary to fortify some cell closer to the fire, that is not part of the final fire border, to delay the fire, such that the firefighters have enough time to fortify the actual protective set.
Note however, that the results we obtained for the static versions in addition heavily utilize topological properties of the fire border in the underlying graph and set of village cells and require restrictions on the possible initial energy and resistance values.
A plethora of problem variations also arises by varying the exact parameters of protection. The firefighters might be allowed to affect only resistance, only energy or both. In the last case, there might be separate budgets for both types of actions; or a joined one, but affected at different weights by the two types of action. For example, preventative measures (i. e., increasing resistance) might be cheaper than actually extinguishing a fire (i. e., decreasing the energy of burning cells).
Besides adapting the exact parameters, more specific variations can also be obtained by focusing on certain graph classes, or simplifying conditions on the initial resistance and energy functions.
Based on the complexity of Hartnell’s Firefighter Problem, as well as the challenges encountered in the static version of the village protection problems, the complexity of these types of dynamic firefighting problems might depend heavily on exploiting some regularity in the input instance.
Chapter 3
Continuous Firefighting
In this chapter we study the effect of delaying barriers in continuous firefighting.
We begin by introducing the canonical problem variant as introduced by Bressan in more detail.
We shortly present both the best known upper and lower bound on the necessary building speedvand argue how the lower bound proof motivates the investigation of delaying barriers. We then present three smaller related problems, in which delaying barriers play three distinct roles.
Afterwards we present the problem variant of continuous firefighting considered in detail in this chapter - to contain the fire in a half-plane. We present some structural observations on optimal strategies that admit some basic lower bounds on the building speed v and subsequently refine these using three different approaches resulting in three different lower bounds, the best of which showsv≥1.66 is necessary. We present an approach to construct a working strategy forv<2 and subsequently optimize this approach to create a strategy for which a speed ofv>1.8772 suffices.
Finally, we discuss some possible extension of our results including some open questions.