## Section: Scientific Foundations

### Scientific Foundations

The project develops tools and theory in the following domains: Discrete Mathematics (in particular Graph Theory), Algorithmics, Combinatorial optimization and Simulation.

Typically, a telecommunication network (or an interconnection
network) is modeled by a graph. A vertex may represent either a
processor or a router or any of the following: a switch, a radio
device, a site or a person. An edge (or arc) corresponds to a
connection between the elements represented by the vertices (logical
or physical connection). We can associate more information both to
the vertices (for example what kind of switch is used, optical or
not, number of ports, equipment cost) and to the edges (weights
which might correspond to length, cost, bandwidth, capacity) or
colors (modeling either wavelengths or frequencies or failures) etc.
Depending on the application, various models can be defined and have
to be specified. This modeling part is an important task. To solve
the problems, we manage, when possible, to find polynomial
algorithms. For example, a maximum set of disjoint paths between two
given vertices is by Menger's theorem equal to the minimum
cardinality of a cut. This problem can be solved in polynomial time
using graph theoretic tools or flow theory or linear programming. On
the contrary, determining whether in a directed graph there exists a
pair of disjoint paths, one from s_{1} to t_{1} and the other from
s_{2} to t_{2} , is an NP-complete problem, and so are all the
problems which aim at minimizing the cost of a network which can
satisfy certain traffic requirements. In addition to deterministic
hypothesis (for example if a connection fails it is considered as
definitely down and not intermittently), the project started
recently to consider probabilistic ones.

Graph coloring is an example of tool which appears in various contexts: WDM networks where colors represent wavelengths, radio networks where colors represent frequencies, fault tolerance where colors represent shared risk resource groups, and scheduling problems. Another tool concerns the development of new algorithmic aspects like parametrized algorithms. A school has been organized on this topic and the research will be conducted under the ANR project AGAPE.

Theoretical results are described after, with more emphasis on those of Graph Theory (Section 6.5 ) and algorithmic aspects (Section 6.6 ).