Synopsis
Exploring why some polyhedral containment problems defy easy answers.
This work examines how certain seemingly straightforward questions about placing one polyhedron inside another fall into the hard, NP-complete camp, shaping our understanding of computational limits in geometry and optimization.
The discussion centers on three set containment problems, each formed by X and Y as special polyhedral or ball-like shapes. It also looks at the related integer containment problem and the complexity of finding integer points inside polyhedra. Readers will see how classical reductions connect these geometric questions to well-known hard problems, and how polynomial-time solvability in some cases does not extend to all.- The problems examined include HB, HW, and BW, which ask whether one set is contained in another under different geometric constraints.
- The work shows these three problems are NP-complete, strengthening the view that some polyhedral containment tasks resist efficient solutions.
- It presents a chain of reductions from the 0-1 knapsack problem and the integer containment problem, illustrating why certain forms become intractable.
- It also discusses ICP2, the problem of finding an integer point in a polyhedron, and establishes its NP-completeness.
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On the Complexity of Four Polyhedral Set Containment Problems (Classic Reprint)
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