## Section: New Results

### Large scale complex structure optimization

**New decomposition methods for the time-dependent combined network design and routing problem:** A significant amount of work has been focussed on the design of telecommunication networks. The performance of different Integer Programming models for various situations has been computationally assessed. One of the settings that has been thoroughly analyzed is a variant where routing decisions (for time-dependent traffic demand), and network design, are combined in a single optimization model. Solving this model with a state-of-the-art solver on representative network topologies, shows that this model quickly becomes intractable. With an extended formulation, both the number of continuous flow variables and the number of fixed charge capacity constraints are multiplied by a factor $\left|V\right|$ (where $V$ represents the set of nodes) leading to large model. However, the linear relaxation of this extended formulation yields much better lower bounds. Nevertheless, even if the extended model provides stronger lower bounds than the aggregated formulation, it suffers from its huge size: solving the linear relaxation of the problem quickly becomes intractable when the network size increases, making the linear relaxation expensive to solve. This observation motivates the analysis of decomposition methods [30].

**Convex piecewise linear unsplittable multicommodity flow problems**
We studied the multi-commodity flow problem with unsplittable flows, and
piecewise-linear costs on the arcs. They show that this problem is NP-hard when there is more
than one commodity. We propose a new MILP models for this problem, that was compared to two
formulations commonly used in the literature. The computational experiments reveal that the new model
is able to obtain very strong lower bounds, and is very efficient to solve the considered problem [40].

**Tree Reconstruction Problems: **
We studied the problem of reconstructing a tree network by knowing
only its set of terminal nodes and their pairwise distances, so that the
reconstructed network has its total edge weight minimized. This problem
has applications in several areas, namely the inference of phylogenetic
trees and the inference of routing networks topology. Phylogenetic
trees allow the understanding of the evolutionary history of species and
can assist in the development of vaccines and the study of
biodiversity. The knowledge of the routing network topology is the basis
for network tomography algorithms and it is a key strategy to the
development of more sophisticated and ambitious traffic control
protocols and dynamic routing algorithms [31].

**Comparison of formulations and solution methods for the discrete ordered p-median problem: **
We presented several new formulations for the Discrete Ordered Median Problem (DOMP) based
on its similarity with some scheduling problems. Some of the new formulations present a considerably
smaller number of constraints to define the problem with respect to some previously known formulations.
Furthermore, the lower bounds provided by their linear relaxations improve the ones obtained with previous
formulations in the literature even when strengthening is not applied. We also present a polyhedral study
of the assignment polytope of our tightest formulation showing its proximity to the convex hull of the
integer solutions of the problem. Several resolution approaches, among which we mention a branch and cut
algorithm, are compared. Extensive computational results on two families of instances, namely randomly
generated and from Beasley's OR-library, show the power of our methods for solving DOMP [34].

**New models and algorithms for integrated vehicle routing problems**

We address a real-life inventory routing problem, which consists in designing routes and managing the inventories of the customers simultaneously. The problem was introduced during the 2016 ROADEF/EURO challenge. The proposed problem is original and complex for several reasons : the logistic ratio optimization objective, the hourly time-granularity for inventory constraints, the driver/trailer allocation management. Clearly, this problem is an optimization problem with complexe structure, for which we proposed a branch-cut-and-price based method : a cut and-column generation procedure was developed, along with a heuristic pricing algorithm to generate new columns and a heuristic fixing procedure to generate integer solutions. The solution method allowed the team including INOCS members to qualify to the final phase of the ROADEF/EURO challenge 2016 [41].

**Column generation approach for pure parsimony haplotyping: **
The knowledge of nucleotides chains that compose the double DNA chain of an individual has a relevant role in detecting diseases and studying populations. However, determining experimentally
the single nucleotides chains that, paired, form a certain portion of the DNA is expensive
and time-consuming. Mathematical programming approaches have been proposed instead, e.g.
formulating the Haplotype Inference by Pure Parsimony problem (HIPP). Abstractly, we are
given a set of genotypes (strings over a ternary alphabet 0, 1, 2) and we want to determine
the smallest set of haplotypes (binary strings over the set 0, 1) so that each genotype can be
'generated' by some pair of haplotypes, meaning that they are compatible with the genotype
and can fully explain its structure.
In order to deal with larger instances, we proposed a new model involving an exponential number
of variables to be solved via column generation, where variables are dynamically introduced into
the model by iteratively solving a pricing problem. We compared different ways of solving the
pricing problem, based on integer programming, smart enumeration and local search heuristic.
The efficiency of the approach is improved by stabilization and by a heuristic to provide a good
initial solution. Results show that, with respect to the linear relaxations of both the polynomial
and exponential-size models, our approach yields a tighter formulation and outperforms in both
efficiency and effectiveness the previous model for instances with a large number of genotypes [39].