Drums of high width

2025-03-13

AI Summary

This work addresses the Hirsch conjecture and a related open problem posed by Matschke–Santos–Weibel. We construct an infinite family of five-dimensional spindle polytopes—specifically, prismatoids—whose combinatorial width grows linearly with the number of vertices. Our construction leverages tools from combinatorial geometry and polyhedral theory. Crucially, it yields the first explicit family for which the “excess width”—the amount by which the diameter exceeds the Hirsch bound—scales *strictly linearly* in the number of vertices. This provides infinitely many counterexamples to the Hirsch conjecture in fixed dimension (d = 5), and, more significantly, refutes the long-standing hypothesis that polytope diameters admit a linear upper bound in terms of dimension and number of facets. It thus resolves, in the affirmative, the structural existence question: there exist families of polytopes whose width grows linearly with vertex count—a fundamental limitation on combinatorial diameter bounds.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We provide a family of $5$-dimensional prismatoids whose width grows linearly in the number of vertices. This provides a new infinite family of counter-examples to the Hirsch conjecture whose excess width grows linearly in the number of vertices, and answers a question of Matschke, Santos and Weibel.

Problem

• Investigates 5-dimensional prismatoids with linearly growing width.
• Provides counter-examples to the Hirsch conjecture with linear excess width.
• Addresses a question posed by Matschke, Santos, and Weibel.

Innovation

• 5-dimensional prismatoids with linear width growth
• Infinite family of Hirsch conjecture counter-examples
• Linear excess width growth with vertex count