## Section: New Results

### Network Design and Routing

Participants : Philippe Meurdesoif, Pierre Pesneau, François Vanderbeck, Benoit Vignac.

#### Decomposition approaches for telecom network routing problems

In collaboration with B. Jaumard from Concordia University in Quebec, B. Vignac and F. Vanderbeck are studying routing and associated design decisions in backbone optical networks.

To accomodate the increase of traffic in telecommunication networks,
today's optical networks have huge capacity thanks to grooming and
wavelength division multiplexing technologies. The wavelength
bandwidth utilization is increased by packing several requests on
the same wavelength. Moreover, several streams can be multiplexed
on an optical signal, each of them supported by a different
wavelength. However, packing multiple requests together in the same
optical stream requires to convert the signal in the electrical
domain at each aggregation of dissagregation of trafic at an origin,
a destination or a bifurcation node. These conversions requires the
installation of expensive ports. Hence, traffic grooming and routing
decisions along with wavelength assignments must be optimized to
reduce opto-electronic system installation cost. This optimization
problem is known as the *grooming, routing and wavelength
assignment* (GRWA) problem.

Given a *physical network* , each link can carry a uniform number
of wavelengths. Each wavelength has the same transport
capacity. Trafic demand over the network take the form of bandwidth
requests defined by their origin and destination and their capacity
requirement that is selected from a discrete set of standard
granularities, {1, 3, 12, 48} that are dividers of the
wavelength capacity. A request must be single path routed. Its route
is defined by a sequence of *optical hops* , each of which is
defined by a path in the physical network along which the signal
remains into the optical domain with no electrical conversion at
intermediate nodes. Traffic routing consists in defining an *optical path* defined by a sequence of optical hops.

We deal with backbone optical network with relatively few nodes (around 20) but thousands of requests. Such instances cannot be dealt with a traditional multi-commodity network flow approach. Instead, we develop and compare several decomposition approaches [41] , [40] , [39] (hierarchical versus nested decomposition): column generation is used to solve the LP relaxation of our models and a rounding procedure provides primal solutions; the LP dual bounds are improved using a cutting plane procedure. In ongoing work, we develop a direct branch-and-cut approach on a pseudo-polynomial formulation, for comparison.

We also studied the impact of imposing a restriction on the number of optical hops in any request route. Indeed, electrical convertions may cause important end-to-end delays. To limit such delay we evaluate different bounds on the number of hops in an optical path. Our study [39] shows that limiting path to 1-hop is very restrictive, while restricting the number of hops to 2, has a very limited impact on the design cost; if we use 3 or more hops, the cost decrease is marginal.

#### Time-dependent formulations for the vehicle routing problem

In collaboration with Teresa Godinho, Luis Gouveia and José Pires of the University of Lisbon and Thomas L. Magnanti of the School of Engineering of the MIT, Pierre Pesneau has studied several time dependent formulations for the unit demand vehicle routing problem [88] . They gave new bounding flow inequalities for a single commodity flow formulation of the problem. They described their impact by projecting on some other sets of variables, such as variables issued of the Picard and Queyranne formulation or the natural set of design variables.

Following up on this work, they proved that some new inequalities obtained by projection are facet defining for the polytope associated with the problem. We are now running more numerical experiments in order to validate in practice the efficiency of our theoretical results.

#### Asymmetric Traveling Salesman Problem

Along with Laurent Alfandari, Sylvie Borne and Lucas Létocart from the University Paris 13, Pierre Pesneau is currently starting a project granted by the working group on operations research (GDR RO) of the CNRS on the study of integer quadratic or integer linear programming formulation for some variants of the Asymmetric Traveling Salesman Problem.

In particular we are studying the case where the distance (in number of links) between the cities of some subset has to be lower bounded. Such case appears, for instance, in the design the shortest circuit for a traveling salesman who has to visit a minimum number of clients before having a break (or a night). Another close application can be found in [91] where the authors consider lower bound on the capacity for a vehicule routing problem.

We have already proposed several formulations of different types for the considered problem. The first two are compact formulations (polynomial number of variables and constraints). Two more formulation will contains a exponential number of constraints leading to the future development of branch-and-cut algorithms. One of these formulation contains an additional set of variables to model the bounding condition in a compact way. We have shown that this formulation is at least as god as the other one. Finally, several formulations where given with a exponential set of variables and for some them, an exponential set of constraints, leading to the conception of branch-and-price(-and cut) algorithms. We plan to compare theoreticaly and practicaly these formulations.