Finding Short Paths on Simple Polytopes

2026-03-05

AI Summary

This study investigates the computational complexity of three fundamental problems on simple polyhedra: shortest monotone paths, shortest pivot sequences for the simplex method, and polyhedral diameter. By integrating techniques from complexity theory, combinatorial analysis of polyhedral structures, and extended formulation constructions, the work establishes that all three problems are NP-hard. This resolves two long-standing open questions: one posed by De Loera et al. (2022) concerning shortest monotone paths and another by Kaibel and Pfetsch (2003) regarding polyhedral diameter. Furthermore, the project constructs extended formulations of polynomial size that enable the identification of feasible paths of linear length in polynomial time, thereby significantly improving the efficiency of path-finding algorithms on polyhedra.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We prove that computing a shortest monotone path to the optimum of a linear program over a simple polytope is NP-hard, thus resolving a 2022 open question of De Loera, Kafer, and Sanit\`a. As a consequence, finding a shortest sequence of pivots to an optimal basis with the simplex method is NP-hard. In fact, we show this is NP-hard already for fractional knapsack polytopes. By applying an additional polyhedral construction, we show that computing the diameter of a simple polytope is NP-hard, resolving a 2003 open problem by Kaibel and Pfetsch. Finally, on the positive side we show that every polytope has a small, simple extended formulation for which a linear length path may be found between any pair of vertices in polynomial time building upon a result of Kaibel and Kukharenko.

Problem

• shortest path
• simple polytope
• linear programming
• simplex method
• polytope diameter

Innovation

• simple polytope
• shortest monotone path
• NP-hardness
• simplex method
• extended formulation