Computing linear extensions for polynomial posets subject to algebraic constraints

TL;DR

This paper introduces an algorithm that efficiently computes specific linear extensions of polynomial-based posets, relevant in gene regulatory network analysis, by leveraging algebraic structures and linear programming.

Contribution

It presents a novel approach to compute admissible linear extensions of polynomial posets using linear programming, tailored for applications in biology.

Findings

Algorithm efficiently computes admissible linear extensions.

Applicable to polynomial posets in gene regulatory networks.

Handles multilinear polynomials through algebraic embedding.

Abstract

In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small "admissible" subset of these linear extensions, determined implicitly by the evaluation map, are of interest. This seemingly novel problem arises in the study of global dynamics of gene regulatory networks in which case the poset is a Boolean lattice. We provide an algorithm for solving this problem using linear programming for arbitrary partial orders of linear polynomials. This algorithm exploits this additional algebraic structure inherited from the polynomials to efficiently compute the admissible linear extensions. The biologically relevant problem involves multilinear polynomials and we provide a construction for embedding it into an instance of the linear problem.