## Section: New Results

### Integrating graph theory and mathematical programming approaches

#### 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.

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