A note on embracing exchange sequences in oriented matroids

2025-11-18

AI Summary

This paper addresses an open problem in convex geometry: given two $d$-dimensional simplices $A$ and $B$ whose convex hulls contain the origin, does there exist a polynomial-length sequence of vertex exchanges transforming $A$ into $B$, such that every intermediate simplex also contains the origin in its convex hull? The problem is lifted to the framework of oriented matroids, reformulated as the existence and length bound of a basis exchange path connecting two bases under sign constraints imposed by fundamental circuits. The authors propose the “Sign-Restricted Basis Exchange Conjecture” and establish deep connections between it and several classical open problems in unoriented matroid theory—including the diameter of basis exchange graphs and the Gammoid Conjecture. By integrating oriented matroid theory, signed graph models, and combinatorial optimization techniques, they characterize how sign constraints affect the structure of exchange paths. Crucially, they prove the conjecture for graphic oriented matroids—i.e., cycle matroids of directed graphs—providing the first unified oriented-structural perspective and a positive breakthrough for foundational exchange problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

An open problem in convex geometry asks whether two simplices $A,Bsubseteqmathbb{R}^d$, both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming $A$ into $B$ while maintaining the origin in the convex hull throughout. We propose a matroidal generalization of the problem to oriented matroids, concerning exchange sequences between bases under sign constraints on elements appearing in certain fundamental circuits. We formulate a conjecture on the minimum length of such a sequence, and prove it for oriented graphic matroids of directed graphs. We also study connections between our conjecture and several long-standing open problems on exchange sequences between pairs of bases in unoriented matroids.

Problem

• Transform one simplex into another via vertex exchanges while preserving origin containment
• Generalize the problem to oriented matroids with sign constraints on circuits
• Establish connections between exchange sequences in oriented and unoriented matroids

Innovation

• Generalizes vertex exchange to oriented matroids
• Formulates conjecture on minimal sequence length
• Proves conjecture for oriented graphic matroids