Propriétés géométriques du nombre chromatique : polyèdres, structures et algorithmes. (Geometric properties of the chromatic number : polyhedra, structure and algorithms)
نویسنده
چکیده
Computing the chromatic number and finding an optimal coloring of a perfect graph can be done efficiently, whereas it is an NP-hard problem in general. Furthermore, testing perfection can be carriedout in polynomial-time. Perfect graphs are characterized by a minimal structure of their stable set polytope: the non-trivial facets are defined by clique inequalities only. Conversely, does a similar facet-structure for the stable set polytope imply nice combinatorial and algorithmic properties of the graph ? A graph is h-perfect if its stable set polytope is completely described by non-negativity, clique and odd-circuit inequalities. Statements analogous to the results on perfection are far from being understood for h-perfection, and negative results are missing. For example, testing h-perfection and determining the chromatic number of an h-perfect graph are unsolved. Besides, no upper bound is known on the gap between the chromatic and clique numbers of an h-perfect graph. Our first main result states that h-perfection is closed under the operations of t-minors (this is a non-trivial extension of a result of Gerards and Shepherd on t-perfect graphs). We also show that the Integer Decomposition Property of the stable set polytope is closed under these operations, and use this to answer a question of Shepherd on 3-colorable h-perfect graphs in the negative. The study of minimally h-imperfect graphs with respect to t-minors may yield a combinatorial co-NP characterization of h-perfection. We review the currently known examples of such graphs, study their stable set polytope and state several conjectures on their structure. On the other hand, we show that the (weighted) chromatic number of certain h-perfect graphs can be obtained efficiently by roundingup its fractional relaxation. This is related to conjectures of Goldberg and Seymour on edge-colorings. Finally, we introduce a new parameter on the complexity of the matching polytope and use it to give an efficient and elementary algorithm for testing h-perfection in line-graphs.
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