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Valid Inequalities, Cutting Planes, and Integrality of the Knapsack Polytope

In: Separable Optimization

Author

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  • Stefan M. Stefanov

    (South-West University Neofit Rilski)

Abstract

In this chapter, the so-called knapsack problem and knapsack polytopeKnapsackpolytope, defined by linear inequality/inequalities and bounds on the variables, are considered. Some important concepts and preliminaries are given at the beginning. As we have observed, knapsack polytope is a feasible region of some problems considered in Part One and Part Two of this book. Some results connected with the generation of valid and dominating inequalities (including the modular arithmetic approach) and cutting planes are presented. Necessary and sufficient conditions are also proved for the knapsack polytope to be integral. Such integrality criteria are useful because the optimal solution to a linear programming problem, when it is solvable, is attained at a vertex of the feasible polytope/polyhedron, and in the case of integrality of the knapsack polytope, the integer problem and the corresponding continuous problem have the same solution(s).

Suggested Citation

  • Stefan M. Stefanov, 2021. "Valid Inequalities, Cutting Planes, and Integrality of the Knapsack Polytope," Springer Optimization and Its Applications, in: Separable Optimization, edition 2, chapter 0, pages 265-280, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-78401-0_14
    DOI: 10.1007/978-3-030-78401-0_14
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