We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to $$\Sigma ^0_2$$
Σ
2
0
-induction over $$\mathsf {RCA}_0$$
RCA
0
. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees $${\mathsf {T}\,}{\mathsf {T}}^1$$
T
T
1
with an extra condition on the solution tree.
In this paper, we consider the problem of flying over an area affected by a natural disaster (e.g. an earthquake) with a fleet of self-piloting unmanned aerial vehicles with cameras or other kinds of sensors on board; the aim is to acquire knowledge of the situation before rescuers start working. We model this situation as a new graph theoretical problem; then, we study its complexity providing an approximation ratio that becomes constant in some special (though practically reasonable) cases; finally, we put in relation the approximability of this new problem and of a well-known one. To the best of our knowledge, no previous work has ever considered all together the constraints we take into account from a theoretical point of view, so this is the first very general graph theoretical model for this problem.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.