Section: New Results

Clustering problems

Clustering problems, and in particular partitionning problems, are widespread in combinatorial optimization. The goal is to partition a set of items in subset satisfying various constraints such as knapsack constraints, cardinality constraints, connectivity constraints, and so on. Beside the PhD thesis of Jérémy Guillot that aims to develop aggregating techniques to handles large scale instances for partitionning problems, the team also study some particular versions.

In [15] we present the application of branch-and-price approaches to the automatic version of the Software Clustering Problem. To tackle this problem, we apply the Dantzig-Wolfe decomposition to a formulation from literature. Given this, we present two Column Generation (CG) approaches to solve the linear programming relaxation of the resulting reformulation: the standard CG approach, and a new approach, which we call Staged Column Generation (SCG). Also, we propose a modification to the pricing subproblem that allows to add multiple columns at each iteration of the CG. We test our algorithms in a set of 45 instances from the literature. The proposed approaches were able to improve the literature results solving all these instances to optimality. Furthermore, the SCG approach presented a considerable performance improvement regarding computational time, number of iterations and generated columns when compared with the standard CG as the size of the instances grows.

In collaboration with researchers from University Paris 6 and Paris 13, we also study the problem of partitionning a geographical area in connected parcels. A first step of this study was to cut the area in two connected parcels while minimizing the dissimilarities inside each parcels. Such partitionning is also called a bond. It happens that in series-parallel garph, a bond correspond to a circuit in the dual graph. In [12] , we give a full description of the circuit polytope on series–parallel graphs. We first show the existence of a compact extended formulation. Though not being explicit, its construction process helps us to inductively provide the description in the original space. As a consequence, using the link between bonds and circuits in planar graphs, we also describe the bond polytope on series–parallel graphs.