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.
Highlights
A great contribution to the development of Computational Complexity Theory for continuous and discrete optimization problems was made by N
Justification that the simplest Shor’s bound coincides with α(G, w) for bipartite and perfect graphs can be made more clear. It is because the quadratic constraints (9)-(10) result in a family of linear inequalities 0 ≤ xi ≤ 1 ∀i ∈ V (G), (11)
W ST AB(G) = {x ∈ R|V | : x satisfies (11), (12), (18), and (19)}. It is associated with the bound αW∗ (G, w), which can be found in polynomial time for an arbitrary graph G [2]
Summary
A great contribution to the development of Computational Complexity Theory for continuous and discrete optimization problems was made by N. Z. Shor, who proposed a technique of Lagrangian (dual) bounds on the global extremum in non-convex quadratic problems [1]. Shor, who proposed a technique of Lagrangian (dual) bounds on the global extremum in non-convex quadratic problems [1] In minimization problems, these will be lower bounds while in maximization problems, these will be upper bounds. These will be lower bounds while in maximization problems, these will be upper bounds This technique includes algorithms for finding Lagrangian bounds based on applying non-differentiable optimization and utilizing functionally redundant constraints to improve the Lagrangian bounds’ accuracy. The Maximum Independent Set problem will be considered in detail, and the difficulty of its solving will be highlighted It will be followed by presenting an improved Shor’s bound for bipartite and perfect graphs and establishing its connection with the Lovasz numbers.
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