Konstantinos Georgiou’s research while affiliated with University of Toronto and other places

We study a primitive vehicle routing-type problem in which a fleet of nunit speed robots start from a point within a non-obtuse triangle $\Delta$, where $n \in \{1,2,3\}$. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing $\Delta$into regions with respect to the type of optimal trajectory that each point P admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan $R_n(P)$ is determined, for $n\in \{1,2,3\}$. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define $R_{n,m} (\Delta)= \max_{P \in \Delta} R_n(P)/R_m(P)$, and we prove that, over all non-obtuse triangles $\Delta$: (i) $R_{1,3}(\Delta)$ ranges from $\sqrt{10}$ to 4, (ii) $R_{2,3}(\Delta)$ ranges from $\sqrt{2}$ to 2, and (iii) $R_{1,2}(\Delta)$ ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle $\Delta$ that maximize $R_{n,m} (\Delta)$, as well as we identify the triangles that determine all $\inf_\Delta R_{n,m}(\Delta)$ and $\sup_\Delta R_{n,m}(\Delta)$ over the set of non-obtuse triangles. Comment: 47 pages, 27 figures

We consider n unit-speed mobile agents initially positioned at the center of a unit disk, tasked with inspecting all points on the disk's perimeter. A perimeter point is considered covered if an agent positioned outside the disk's interior has unobstructed visibility of it, treating the disk itself as an obstacle. For n=1, this problem is referred to as the shoreline problem with a known distance. Isbell in 1957 derived an optimal trajectory that minimizes the worst-case inspection time for that problem. The one-agent version of the problem was originally proposed as a more tractable variant of Bellman's famous lost-in-the-forest problem. Our contributions are threefold. First, and as a warm-up, we extend Isbell's findings by deriving worst-case optimal trajectories addressing the partial inspection of a section of the disk, hence deriving an alternative proof of optimality for inspecting the disk with $n \geq 2$ agents. Second, we analyze the average-case inspection time, assuming a uniform distribution of perimeter points (equivalent to randomized inspection algorithms). Using spatial discretization and Nonlinear Programming (NLP), we propose feasible solutions to the continuous problem and evaluate their effectiveness compared to NLP solutions. Third, we establish Pareto-optimal bounds for the multi-objective problem of jointly minimizing the worst-case and average-case inspection times.

We consider \emph{weighted group search on a disk}, which is a search-type problem involving 2 mobile agents with unit-speed. The two agents start collocated and their goal is to reach a (hidden) target at an unknown location and a known distance of exactly 1 (i.e., the search domain is the unit disk). The agents operate in the so-called \emph{wireless} model that allows them instantaneous knowledge of each others findings. The termination cost of agents' trajectories is the worst-case \emph{arithmetic weighted average}, which we quantify by parameter w, of the times it takes each agent to reach the target, hence the name of the problem. Our work follows a long line of research in search and evacuation, but quite importantly it is a variation and extension of two well-studied problems, respectively. The known variant is the one in which the search domain is the line, and for which an optimal solution is known. Our problem is also the extension of the so-called \emph{priority evacuation}, which we obtain by setting the weight parameter w to 0. For the latter problem the best upper/lower bound gap known is significant. Our contributions for weighted group search on a disk are threefold. \textit{First}, we derive upper bounds for the entire spectrum of weighted averages w. Our algorithms are obtained as a adaptations of known techniques, however the analysis is much more technical. \textit{Second}, our main contribution is the derivation of lower bounds for all weighted averages. This follows from a \emph{novel framework} for proving lower bounds for combinatorial search problems based on linear programming and inspired by metric embedding relaxations. \textit{Third}, we apply our framework to the priority evacuation problem, improving the previously best lower bound known from 4.38962 to 4.56798, thus reducing the upper/lower bound gap from 0.42892 to 0.25056.

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.

p> We initiate the study of a problem on searching and fetching, motivated by real-life surveillance and search-and-rescue operations where unmanned vehicles, e.g. drones, search for victims in areas of a disaster. In treasure-evacuation, we are interested in designing algorithms that minimize the time it takes for a treasure (a victim) to be discovered and brought (fetched) to the exit (shelter) by any of two robots (rescuers) which are performing in a distributed environment (the case of searching and fetching with 1 robot has been previously considered).The communication protocol between the robots is either wireless, where information is shared at any time, or face-to-face, where information can be shared only if the robots meet. For both models we obtain upper bounds for fetching the treasure to the exit. Our algorithms make explicit use of the distance between the treasure and the exit, which is assumed to be known in advance, showing this way how partial information of the unknown input can be beneficial. Our main technical contribution pertains to the face-to-face model. More specifically, we demonstrate how robots can exchange information without meeting, effectively achieving a highly efficient treasure-evacuation protocol which is minimally affected by the lack of distant communication. Finally, we complement our positive results above by providing a lower bound in the face-to-face model.</p

p>Let G be a graph in which each vertex initially has weight 1. 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 G , denoted by a _u (G) , is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdõs-Rényi random graph process ( G(n,m) ) ^N _m=0 , where N =(ⁿ ₂).We show that asymptotically almost surely a _u (G)(n,m)) = 1 right at the time step the random graph process creates a connected graph. Since trivially a _u (G)(n,m)) ≥ 2 if the graphs is disconnected, the result holds in the strongest possible sense. </p