Оцiнки Шора для зваженого числа стiйкостi графа

Abstract

Application of a technique of dual Lagrangian quadratic bounds of N.Z. Shor to studying the Maximum Weighted Independent Set problem is described. By the technique, two such N.Z. Shor’s upper bounds are obtained. These are bounds of the graph weighted independence number $ \alpha (G, w) $, which can be found in polynomial time. The first bound $ \psi (G, w) $ is associated with a quadratic model of the Maximum Weighted Independent Set problem and coincides with the known Lov\'asz number $ \vartheta (G, w) $. The second bound $ \psi_1 (G, w) $ corresponds to the same quadratic model supplemented by a family of functionally redundant quadratic constraints and is able to improve the accuracy of the upper bound $ \alpha (G, w) $ for special graph families. It is shown that, if graph is bipartite or perfect, $ \psi (G, w)= \alpha (G, w) $, while $ \psi_1 (G, w) =\alpha (G, w) $ for $ t $- or $ W_p $-perfect graphs. Based on the graph classes that were singled out, a technique is demonstrated, which enables us to form new classes of graphs for which polynomial solvability of the Maximum Weighted Independent Set problem is preserved. Thus, by an example of the Maximum Weighted Independent Set problem in a graph, it is shown how the Lagrangian bounds’ technique can be applied to solving an issue of single outing new classes of polynomial solvable combinatorial optimization problems. This approach can be used for improving known bounds of the objective function in combinatorial optimization problems as well as for justifying their polynomial solvability.