Faster algorithms for circuits in the Cayley-Menger algebraic matroid

TL;DR

This paper introduces an improved algebraic-combinatorial algorithm for computing circuit polynomials in the Cayley-Menger algebraic matroid, significantly enhancing efficiency for complex Distance Geometry problems.

Contribution

It extends previous algorithms by leveraging additional algebraic structure, enabling faster calculations of circuit polynomials in the Cayley-Menger ideal.

Findings

Successfully computed a complex circuit polynomial in 30 minutes that previously took days

Enhanced the algorithm to utilize non-circuit generators and irreducible polynomials

Implemented the method in Mathematica, enabling practical applications in Distance Geometry

Abstract

A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths. For minimally rigid graphs, the double-exponential Gr\"obner Bases algorithm with an elimination order can be applied, in theory, but it is impractical even for small instances. By relating the problem to the computation of circuit polynomials in the Cayley-Menger ideal, we recently proposed an algebraic-combinatorial approach and an elimination algorithm for circuit polynomials [23]. It is guided by a tree structure whose leaves correspond to complete $K_4$ graphs and whose nodes perform algebraic resultant operations. In this paper we uncover further combinatorial structure in the Cayley-Menger algebraic matroid that leads to an extension of our algorithm. In particular, we generalize the combinatorial resultant operation of [23] to take advantage of the non-circuit generators and irreducible polynomials in the Cayley-Menger ideal and use them as leaves of the tree guiding the elimination. Our new method has been implemented in Mathematica and allows previously unobtainable calculations to be carried out. In particular, the $K_{3,3}$-plus-one circuit polynomial, with over one million terms in 10 variables and whose calculation crashed after several days with the previous method of [23], succeeded now in approx. 30 minutes.