Yann Disser

Reconstructing visibility graphs with simple robots

We consider the problem of finding a minimalistic configuration of sensors that enable a simple robot inside an initially unknown polygon P on n vertices to reconstruct the visibility graph of P. The robot can sense features of its environment through its sensors, and it is allowed to move from vertex to vertex. We aim at understanding which sensorial capabilities are sufficient for the reconstruction of the visibility graph of P. We are able to show that the combinatorial visibilities at every vertex do not contain enough information even when combined with the knowledge of the exact interior angle at each vertex. Using sensors that can put distant vertices into a spatial relation on the other hand can in some cases enable our robot to reconstruct the visibility graph of P. We show that this is true for a sensor that can distinguish whether the angle between the lines toward two visible vertices is convex or reflex, as long as the robot is capable of identifying the vertex it last visited. We also show that measuring angles exactly is enough, if the robot has a compass.

Improving the H k -bound on the price of stability in undirected Shapley network design games

In this article we show that the price of stability of Shapley network design games on undirected graphs with k players is at most k 3 ( k + 1 ) / 2 - k 2 1 + k 3 ( k + 1 ) / 2 - k 2 H k = ( 1 - ? ( 1 / k 4 ) ) H k , where H k denotes the k-th harmonic number. This improves on the known upper bound of H k , which is also valid for directed graphs but for these, in contrast, is tight. Hence, we give the first non-trivial upper bound on the price of stability for undirected Shapley network design games that is valid for an arbitrary number of players. Our bound is proved by analyzing the price of stability restricted to Nash equilibria that minimize the potential function of the game. We also present a game with k = 3 players in which such a restricted price of stability is 1.634. This shows that the analysis of Bilo and Bove (2011) 3 is tight. In addition, we give an example for three players that improves the lower bound on the (unrestricted) price of stability to 1.571.

Fast collaborative graph exploration

We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure that every vertex has been visited by some agent. We consider two communication models: one in which all agents have global knowledge of the state of the exploration, and one in which agents may only exchange information when simultaneously located at the same vertex. As our main result, we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most D from r in time O(D), using a team of agents of polynomial size k=Dn1+e 0. Our strategy works in the local communication model, without knowledge of global parameters such as n or D. We also obtain almost-tight bounds on the asymptotic relation between exploration time and team size, for large k. For any constant c>1, we show that in the global communication model, a team of k=Dnc agents can always complete exploration in

Reconstructing a simple polygon from its angles

D(1+ \frac{1}{c-1} +o(1))

Mapping a polygon with holes using a compass

time steps, whereas at least

Scheduling Bidirectional Traffic on a Path

D(1+ \frac{1}{c} -o(1))

Packing a Knapsack of Unknown Capacity

steps are sometimes required. In the local communication model,

Polygon-Constrained Motion Planning Problems

D(1+ \frac{2}{c-1} +o(1))

Telling convex from reflex allows to map a polygon

steps always suffice to complete exploration, and at least

Energy-Efficient Delivery by Heterogeneous Mobile Agents.

D(1+ \frac{2}{c} -o(1))