16 research outputs found
Rendezvous of Heterogeneous Mobile Agents in Edge-weighted Networks
We introduce a variant of the deterministic rendezvous problem for a pair of
heterogeneous agents operating in an undirected graph, which differ in the time
they require to traverse particular edges of the graph. Each agent knows the
complete topology of the graph and the initial positions of both agents. The
agent also knows its own traversal times for all of the edges of the graph, but
is unaware of the corresponding traversal times for the other agent. The goal
of the agents is to meet on an edge or a node of the graph. In this scenario,
we study the time required by the agents to meet, compared to the meeting time
in the offline scenario in which the agents have complete knowledge
about each others speed characteristics. When no additional assumptions are
made, we show that rendezvous in our model can be achieved after time in a -node graph, and that such time is essentially in some cases
the best possible. However, we prove that the rendezvous time can be reduced to
when the agents are allowed to exchange bits of
information at the start of the rendezvous process. We then show that under
some natural assumption about the traversal times of edges, the hardness of the
heterogeneous rendezvous problem can be substantially decreased, both in terms
of time required for rendezvous without communication, and the communication
complexity of achieving rendezvous in time
When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots
A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots move perpetually along the domain, represented by all points belonging to the graph edges, without exceeding their maximum speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e., they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolman? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by Ifk(G) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing Ifk(G). We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness Ifk(G)=(f+1)|E|/k, where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphsâa known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs
The Fagnano Triangle Patrolling Problem
We investigate a combinatorial optimization problem that involves patrolling
the edges of an acute triangle using a unit-speed agent. The goal is to
minimize the maximum (1-gap) idle time of any edge, which is defined as the
time gap between consecutive visits to that edge. This problem has roots in a
centuries-old optimization problem posed by Fagnano in 1775, who sought to
determine the inscribed triangle of an acute triangle with the minimum
perimeter. It is well-known that the orthic triangle, giving rise to a periodic
and cyclic trajectory obeying the laws of geometric optics, is the optimal
solution to Fagnano's problem. Such trajectories are known as Fagnano orbits,
or more generally as billiard trajectories. We demonstrate that the orthic
triangle is also an optimal solution to the patrolling problem.
Our main contributions pertain to new connections between billiard
trajectories and optimal patrolling schedules in combinatorial optimization. In
particular, as an artifact of our arguments, we introduce a novel 2-gap
patrolling problem that seeks to minimize the visitation time of objects every
three visits. We prove that there exist infinitely many well-structured
billiard-type optimal trajectories for this problem, including the orthic
trajectory, which has the special property of minimizing the visitation time
gap between any two consecutively visited edges. Complementary to that, we also
examine the cost of dynamic, sub-optimal trajectories to the 1-gap patrolling
optimization problem. These trajectories result from a greedy algorithm and can
be implemented by a computationally primitive mobile agent