Section: New Results
Integrating graph theory and mathematical programming approaches
Participants : Cédric Joncour, Philippe Meurdesoif, Arnaud Pêcher, Gautier Stauffer, François Vanderbeck, Annegret Wagler.
Circular chromatic number
A major result of graph theory is that the chromatic number of a perfect graph is computable in polynomial time (Grötschel, Lovász and Schrijver 1981). The circular chromatic number is a well-studied refinement of the chromatic number of a graph. Xuding Zhu noticed in 2000 that circular cliques are the relevant circular counterpart of cliques, with respect to the circular chromatic number, thereby introducing circular-perfect graphs, a super-class of perfect graphs. It is unknown whether the circular-chromatic number of a circular-perfect graph is computable in polynomial time in general. This is an important issue. Indeed, since the chromatic number of a graph is the integer ceiling of its circular-chromatic number, if the circular chromatic number of a circular-perfect graph is computable in polynomial time then it would imply Grötschel, Lovász and Schrijver's result.
In [20] , we design a polynomial time algorithm that computes this circular chromatic number when the circular-perfect graphs is claw-free. In 2005, Coulonges, Pêcher and Wagler, introduced the intermediate class of strongly circular-perfect graphs, as those circular-perfect graphs whose complements are also circular-perfect. For the triangle free cases, we managed to fully characterize these graphs, and gave a polynomial time algorithm to recognize them [15] . Coulonges, Pêcher and Wagler also introduced a -perfect graphs, another super-class of perfect graphs. We bounded their imperfection ratio by 1.5 [16] .
We have also introduced the circular-clique polytope, and used it to prove that the weighted circular-clique number, and thus the circular chromatic number, of a strongly circular-perfect graph which is also a -perfect is computable in polynomial time. We also proved that the circular chromatic number of a strongly circular-perfect graph is computable in polynomial time. Unexpectedly, we also used the circular-cliques polytope to prove that the circular stability number, and thus the stability number, of a fuzzy circular-interval graph is computable in polynomial time [23] .
Stable set problem and polytope in claw-free graphs
Our focus is now two-fold : first we are exploiting our proof of the Ben Rebea conjecture further to give a clearer picture of the polyhedral nature for quasi-line graphs, in particular we want to describe precisely the non rank facets of quasi-line graphs and along the way, to prove a conjecture of Pecher and Wagler [139] for webs (a subclass). In a more general scope, A. Pêcher and A. Wagler describe in [19] new facets for the stable set polytope of claw-free graphs. Second, our current algorithmic developments suggest that we can provide an extended formulation of the problem and therefore an extended characterization of the stable set polytope for claw-free graphs, this would partially answer a 30 years old question. Moreover, we envision that the technique we are developing has a broader spectrum of application than the stable set problem in claw-free graphs. We are currently setting the foundations for a more general setting. This work is a collaboration with University of Tor Vergata, Roma, Italy (Prof. Gianpaolo Oriolo, Yuri Fuenza). A preliminary result has been submitted to the IPCO 2010 conference.
Using graph theory for solving orthogonal knapsack problems
With C. Joncour (PhD student), we investigate the orthogonal knapsack problem, with the help of graph theory [106] . Fekete and Schepers managed a recent breakthrough in solving multi-dimensional orthogonal placement problems by using an efficient representation of all geometrically symmetric solutions by a so called packing class involving one interval graph (whose complement admits a transitive orientation: each such orientation of the edges corresponds to a specific placement of the forms) for each dimension. Though Fekete & Schepers' framework is very efficient, we have however identified several weaknesses in their algorithms: the most obvious one is that they do not take advantage of the different possibilities to repesent interval graphs.
In [27] , we propose to represent these graphs by matrices with consecutive ones on each row. We proposed a branch and bound algorithm already used to solve the one-dimension knapsack problem. The first results are encouraging as we are able to solve more problems than the algorithm proposed by Fekete & Schepers. The ongoing work is the development of a branch-and-price algorithm for this problem using this way to represent interval graphs.
