Multi-Agent Search-Type Problems on Polygons

We present several advancements in search-type problems for fleets of mobile agents operating in two dimensions under the wireless model. Potential hidden target locations are equidistant from a central point, forming either a disk (infinite possible locations) or regular polygons (finite possible locations). Building on the foundational disk evacuation problem, the disk priority evacuation problem with \(k\) Servants, and the disk \(w\)-weighted search problem, we make improvements on several fronts. First we establish new upper and lower bounds for the \(n\)-gon priority evacuation problem with \(1\) Servant for \(n \leq 13\), and for \(n_k\)-gons with \(k=2, 3, 4\) Servants, where \(n_2 \leq 11\), \(n_3 \leq 9\), and \(n_4 \leq 10\), offering tight or nearly tight bounds. The only previous results known were a tight upper bound for \(k=1\) and \(n=6\) and lower bounds for \(k=1\) and \(n \leq 9\). Second, our work improves the best lower bound known for the disk priority evacuation problem with \(k=1\) Servant from \(4.46798\) to \(4.64666\) and for \(k=2\) Servants from \(3.6307\) to \(3.65332\). Third, we improve the best lower bounds known for the disk \(w\)-weighted group search problem, significantly reducing the gap between the best upper and lower bounds for \(w\) values where the gap was largest. These improvements are based on nearly tight upper and lower bounds for the \(11\)-gon and \(12\)-gon \(w\)-weighted evacuation problems, while previous analyses were limited only to lower bounds and only to \(7\)-gons.