Section: Scientific Foundations
Mathematical Programming and Graph Theory
The relationship between graph theory and mathematical programming has led to several famous research advances. Let us cite just a few landmarks. The matching problem (selecting disjoint edges in a graph) is historically the first integer programming problem that could not be solved by linear programming (no compact ideal formulation being known) but for which an efficient (polynomial time) combinatorial algorithm was known  (ideal formulations were only known for network flow problems at the time). The combinatorial algorithm lead to a polyhedral description of the matching polytope with exponentially many constraints separable in polynomial time  .
The stable set problem in claw-free graphs is a fundamental generalization of the matching problem but while the later has been intensively studied since the seminal work of Edmonds  in the 60's and is now well understood, the stable set problem in claw-free graphs has still many "facets" to reveal. Our long term goal is to develop this theory further to the point were the stable set problem in claw-free graphs becomes as fundamental as the matching in polyhedral combinatorics.
The algorithmic aspect has attracted the attention of many researchers over the last three decades and is quite well understood. Minty  provides the first polynomial time algorithm to solve the problem, then Sbihi  , Lovasz and Plummer  , Nakamura and Tamura  , Schrijver  , Nobili and Sassano  have proposed alternative methods and often improved the complexity. We have recently proposed an alternative algorithm that outperforms all previous ones  .
In term of polyhedral structure, most of the results about the stable set polytope of claw-free graphs were negative and discouraged researchers to work on the topic : as an example, Giles and Trotter  showed that the facets of the polytope could have arbitrary high values and many coefficients. Nevertheless recent decomposition results for claw-free graphs from Chudnovsky and Seymour  attracted new attention to the problem. In particular, building upon their results, we could prove that the so-called Ben Rebea conjecture was true for the stable set polytope of quasi-line graphs (a very interesting transition class between linegraphs - i.e. matching - and claw-free)  .
Thus, the graph theory tools allow to derive better models/formulations for combinatorial optimization problems (f.i. graph theory characterization of forbidden subgraphs can sometimes be directly expressed as constraints in a mathematical program). Moreover, combinatorial procedure from graph theory can also serve as subroutines in mathematical programming approach, f.i., for cut separation or column generation. In particular, on the issue of symmetries, we believe that progress can come from the complementarity between graph theory and mathematical programming. This is illustrated by the work of Fekete and Schepers (2004)  on the 2-dimensional placement problem. In our project, we also play the reverse complementarity, i.e. to use mathematical programming techniques to make progress in graph theory.
In this area, there is potential for collaboration between our team and the INRIA team MASCOTTE, among others.