Study of Problems Involving Mobile Agents

In this thesis we study two different problems that involve the movement of some entities within a well-defined space. One problem is in a continuous domain and the other problem is in the context of discrete spaces of graphs. 

The first problem is a primitive vehicle routing-type problem in which a fleet of n ∈ {1, 2, 3} unit speed robots start from a point within a non-obtuse triangle ∆, where the goal is to design robots’ trajectories to visit all edges of the triangle with the smallest visitation time makespan. Here, makespan is defined as the maximum of the times needed by each robot to finish the work. We begin our study by introducing a framework for subdividing ∆ into regions with respect to the type of optimal trajectories that each starting point P ∈ ∆ admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan Rn(P) is determined, for n ∈ {1, 2, 3}. These subdivisions lead to our main result, which involves makespan trade-offs with respect to the size of the fleet. 

In the next problem we study unit acquisition process in the context of random graphs. Here each vertex of a given graph has unit weight initially. In each step, the unit weight from a vertex u to a neighbouring vertex v can be moved, provided that the weight on v is at least as large as the weight on u. The unit acquisition number of graph G, denoted by au(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process, over all acquisition algorithms. We investigate the Erdõs-Rényi random graph process and show that asymptotically almost surely au(G) = 1 when the random graph process creates a connected graph. The result holds in the strongest possible sense, since au(G) ≥ 2 if the graph is disconnected.