An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting

TL;DR

This paper investigates the realizability of permutation triples as geometric permutations of convex sets in three-dimensional space, transforming a semi-algebraic problem into a combinatorial one and providing an algorithm for analysis.

Contribution

It introduces a novel combinatorial lifting approach to decide geometric permutation realizability, offering new insights and an effective algorithm.

Findings

Decidable criteria for geometric permutation realizability.

Algorithmic framework for analyzing semi-algebraic sets.

Enhanced understanding of the structure of geometric permutations.

Abstract

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in $\mathbb{R}^3$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.

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