25 research outputs found
3/2 Firefighters are not enough
The firefighter problem is a monotone dynamic process in graphs that can be
viewed as modeling the use of a limited supply of vaccinations to stop the
spread of an epidemic. In more detail, a fire spreads through a graph, from
burning vertices to their unprotected neighbors. In every round, a small amount
of unburnt vertices can be protected by firefighters. How many firefighters per
turn, on average, are needed to stop the fire from advancing? We prove tight
lower and upper bounds on the amount of firefighters needed to control a fire
in the Cartesian planar grid and in the strong planar grid, resolving two
conjectures of Ng and Raff.Comment: 8 page
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
A Fire Fighter's Problem
Suppose that a circular fire spreads in the plane at unit speed. A single
fire fighter can build a barrier at speed . How large must be to
ensure that the fire can be contained, and how should the fire fighter proceed?
We contribute two results.
First, we analyze the natural curve \mbox{FF}_v that develops when the
fighter keeps building, at speed , a barrier along the boundary of the
expanding fire. We prove that the behavior of this spiralling curve is governed
by a complex function , where and are real
functions of . For all zeroes are complex conjugate
pairs. If denotes the complex argument of the conjugate pair nearest to
the origin then, by residue calculus, the fire fighter needs
rounds before the fire is contained. As decreases towards these two
zeroes merge into a real one, so that argument goes to~0. Thus, curve
\mbox{FF}_v does not contain the fire if the fighter moves at speed .
(That speed is sufficient for containing the fire has been proposed
before by Bressan et al. [7], who constructed a sequence of logarithmic spiral
segments that stay strictly away from the fire.)
Second, we show that any curve that visits the four coordinate half-axes in
cyclic order, and in inreasing distances from the origin, needs speed
, the golden ratio, in order to contain the fire.
Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower
and upper boundsComment: A preliminary version of the paper was presented at SoCG 201