On the Complexity of Separating Cutting Planes for the Knapsack Polytope
Alberto Del Pia, Jeff Linderoth, Haoran Zhu
TL;DR
This paper proves that separating certain classes of inequalities for the knapsack polytope is NP-complete, but identifies special cases where separation can be done efficiently, advancing understanding of computational complexity in integer programming.
Contribution
The paper establishes NP-completeness for separation problems of extended cover, (1,k)-configuration, and weight inequalities, and identifies polynomial-time solvable cases.
Findings
Separation for extended cover inequalities is NP-complete.
Separation for (1,k)-configuration inequalities is NP-complete.
Separation for weight inequalities is NP-complete.
Abstract
We close three open problems in the separation complexity of valid inequalities for the knapsack polytope. Specifically, we establish that the separation problems for extended cover inequalities, (1,k)-configuration inequalities, and weight inequalities are all NP-complete. We also give a number of special cases where the separation problem can be solved in polynomial time.
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Taxonomy
TopicsOptimization and Packing Problems · Material Properties and Processing · Computational Geometry and Mesh Generation