Subtraction-free complexity, cluster transformations, and spanning trees
Abstract
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev. We develop such algorithms for computing generating functions of spanning trees, both directed and undirected. A comparison to the lower bound due to M. Jerrum and M. Snir shows that in subtraction-free computations, "division can be exponentially powerful." Finally, we give a simple example where the gap between ordinary and subtraction-free complexity is exponential.
Cite
@article{arxiv.1307.8425,
title = {Subtraction-free complexity, cluster transformations, and spanning trees},
author = {Sergey Fomin and Dima Grigoriev and Gleb Koshevoy},
journal= {arXiv preprint arXiv:1307.8425},
year = {2014}
}
Comments
30 pages. Version 4: Section 8 edited. Version 3: Section 8 is new. Version 2: title changed; Section 7 is new; comparison with the Jerrum-Snir lower bound added