In cutting stock problems, after an optimal (minimal stock usage) cutting plan has been devised, one might want to further reduce the operational costs by minimizing the number of setups. A setup operation occurs each time a different cutting pattern begins to be produced. The related optimization problem is known as the Pattern Minimization Problem, and it is particularly hard to solve exactly. In this paper, we present different techniques to strengthen a formulation proposed in the literature. Dual feasible functions are used for the first time to derive valid inequalities from different constraints of the model, and from linear combinations of constraints. A new arc flow formulation is also proposed. This formulation is used to define the branching scheme of our branch-and-price-and-cut algorithm, and it allows the generation of even stronger cuts by combining the branching constraints with other constraints of the model. The computational experiments conducted on instances from the literature show that our algorithm finds optimal integer solutions faster than other approaches. A set of computational results on random instances is also reported.

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities.

We first describe four recent methods to cluster vertices of an undirected non weighted connected graph. They are all based on very different principles. The fifth is a combination of classical ideas in optimization applied to graph partitioning. We compare these methods according to their ability to recover classes initially introduced in random graphs with more edges within the classes than between them.

Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs G where the stable set polytope STAB(G) coincides with the fractional stable set polytope QSTAB(G). For all imperfect graphs G it holds that STAB(G) ⊂ QSTAB(G). It is, therefore, natural to use the difference between the two polytopes in order to decide how far an imperfect graph is away from being perfect. We discuss three different concepts, involving the facet set of STAB(G), the disjunctive index of QSTAB(G), and the dilation ratio of the two polytopes.
Including only certain types of facets for STAB(G), we obtain graphs that are in some sense close to perfect graphs, for example minimally imperfect graphs, and certain other classes of so-called rank-perfect graphs. The imperfection ratio has been introduced by Gerke and McDiarmid [12] as the dilation ratio of STAB(G) and QSTAB(G), whereas Aguilera et al. [1] suggest to take the disjunctive index of QSTAB(G) as the imperfection index of G. For both invariants there exist no general upper bounds, but there are bounds known for the imperfection ratio of several graph classes [7,12].
Outgoing from a graph-theoretical interpretation of the imperfection index, we prove that there exists no upper bound on the imperfection index for those graph classes with a known bounded imperfection ratio. Comparing the two invariants on those classes, it seems that the imperfection index measures imperfection much more roughly than the imperfection ratio; we, therefore, discuss possible directions for refinements.

Wireless LAN using IEEE 802.11 networks are now widely deployed at home by residential users or in hot spots by telecommunication operators. A hot spot is a place where a set of access points (APs) are located nearby each other and can serve many users. Since perturbations can degrade the quality of the signal, a careful channel assignment to each AP has to be done. Channel assignment of APs at hot spots, and more generally setup configuration and management, is still often done manually. In this paper, we consider a modeling that enables optimization of channel assignment with respect to the dynamic behavior of end-users. We prove our problem's formulation to correspond to the Minimum Interference Frequency Assignment Problem, and hence the problem to be NP-hard. We propose and compare three different tabu search methods to solve the problem of channel assignment in 802.11 WLAN networks. The first one, called TabuObj, tackles the problem using directly the objective function associated with the model. The second one, called TabuApproxObj, uses a simplified and approximate objective function in order to visit more solutions during the same amount of time, i.e. to be quicker than TabuObj. The third one, called TabuLevel, is even more quicker and is based on the following philosophy: under time constraints, it could be judicious to explore very quickly lots of solutions, rather than spending much computation time for the evaluation of each solution, and hence only considering a few solutions. Those three methods are then compared based on time constraints and on the quality of their solutions.