24 research outputs found
Time-Varying Graphs and Dynamic Networks
The past few years have seen intensive research efforts carried out in some
apparently unrelated areas of dynamic systems -- delay-tolerant networks,
opportunistic-mobility networks, social networks -- obtaining closely related
insights. Indeed, the concepts discovered in these investigations can be viewed
as parts of the same conceptual universe; and the formal models proposed so far
to express some specific concepts are components of a larger formal description
of this universe. The main contribution of this paper is to integrate the vast
collection of concepts, formalisms, and results found in the literature into a
unified framework, which we call TVG (for time-varying graphs). Using this
framework, it is possible to express directly in the same formalism not only
the concepts common to all those different areas, but also those specific to
each. Based on this definitional work, employing both existing results and
original observations, we present a hierarchical classification of TVGs; each
class corresponds to a significant property examined in the distributed
computing literature. We then examine how TVGs can be used to study the
evolution of network properties, and propose different techniques, depending on
whether the indicators for these properties are a-temporal (as in the majority
of existing studies) or temporal. Finally, we briefly discuss the introduction
of randomness in TVGs.Comment: A short version appeared in ADHOC-NOW'11. This version is to be
published in Internation Journal of Parallel, Emergent and Distributed
System
Temporal Graph Traversals: Definitions, Algorithms, and Applications
A temporal graph is a graph in which connections between vertices are active
at specific times, and such temporal information leads to completely new
patterns and knowledge that are not present in a non-temporal graph. In this
paper, we study traversal problems in a temporal graph. Graph traversals, such
as DFS and BFS, are basic operations for processing and studying a graph. While
both DFS and BFS are well-known simple concepts, it is non-trivial to adopt the
same notions from a non-temporal graph to a temporal graph. We analyze the
difficulties of defining temporal graph traversals and propose new definitions
of DFS and BFS for a temporal graph. We investigate the properties of temporal
DFS and BFS, and propose efficient algorithms with optimal complexity. In
particular, we also study important applications of temporal DFS and BFS. We
verify the efficiency and importance of our graph traversal algorithms in real
world temporal graphs
On the Treewidth of Dynamic Graphs
Dynamic graph theory is a novel, growing area that deals with graphs that
change over time and is of great utility in modelling modern wireless, mobile
and dynamic environments. As a graph evolves, possibly arbitrarily, it is
challenging to identify the graph properties that can be preserved over time
and understand their respective computability.
In this paper we are concerned with the treewidth of dynamic graphs. We focus
on metatheorems, which allow the generation of a series of results based on
general properties of classes of structures. In graph theory two major
metatheorems on treewidth provide complexity classifications by employing
structural graph measures and finite model theory. Courcelle's Theorem gives a
general tractability result for problems expressible in monadic second order
logic on graphs of bounded treewidth, and Frick & Grohe demonstrate a similar
result for first order logic and graphs of bounded local treewidth.
We extend these theorems by showing that dynamic graphs of bounded (local)
treewidth where the length of time over which the graph evolves and is observed
is finite and bounded can be modelled in such a way that the (local) treewidth
of the underlying graph is maintained. We show the application of these results
to problems in dynamic graph theory and dynamic extensions to static problems.
In addition we demonstrate that certain widely used dynamic graph classes
naturally have bounded local treewidth
On the Feasibility of Maintenance Algorithms in Dynamic Graphs
Near ubiquitous mobile computing has led to intense interest in dynamic graph
theory. This provides a new and challenging setting for algorithmics and
complexity theory. For any graph-based problem, the rapid evolution of a
(possibly disconnected) graph over time naturally leads to the important
complexity question: is it better to calculate a new solution from scratch or
to adapt the known solution on the prior graph to quickly provide a solution of
guaranteed quality for the changed graph?
In this paper, we demonstrate that the former is the best approach in some
cases, but that there are cases where the latter is feasible. We prove that,
under certain conditions, hard problems cannot even be approximated in any
reasonable complexity bound --- i.e., even with a large amount of time, having
a solution to a very similar graph does not help in computing a solution to the
current graph. To achieve this, we formalize the idea as a maintenance
algorithm. Using r-Regular Subgraph as the primary example we show that
W[1]-hardness for the parameterized approximation problem implies the
non-existence of a maintenance algorithm for the given approximation ratio.
Conversely we show that Vertex Cover, which is fixed-parameter tractable, has a
2-approximate maintenance algorithm. The implications of NP-hardness and
NPO-hardness are also explored
Puissance de l'attente aux stations pour l'exploration des réseaux de transport public
International audienceNous étudions le problème de l'exploration, par une entité mobile, d'une classe de graphes dynamiques appelés graphes périodiquement variables (PV-graphes). Ils sont définis par un ensemble de transporteurs suivant infiniment leur route respective le long des stations du réseau, et modélisent donc naturellement les réseaux de transport public. Flocchini, Mans et Santoro [FMS09] ont étudié ce problème dans le cas où l'agent doit toujours rester sur les transporteurs. Dans ce papier, nous étudions l'impact de la capacité d'attendre sur les stations. Nous prouvons que l'attente sur les stations permet à l'agent d'atteindre de meilleures complexités en pire cas : le nombre de mouvements est réduit d'un facteur multiplicatif d'au moins Theta(p), et la complexité en temps passe de Theta(kp^2) à Theta(np), où n est le nombre de stations, k le nombre de transporteurs, et p la période maximale (n <= kp dans tout PV-graphe connexe). Par ailleurs, l'algorithme que nous proposons pour prouver les bornes supérieures permet de réaliser la cartographie du PV-graphe, en plus de l'explorer