Subtraction-Free Complexity, Cluster Transformations, and Spanning Trees

• Published in: Foundations of computational mathematics Vol. 16; no. 1; pp. 1 - 31
• Main Authors: Fomin, Sergey, Grigoriev, Dima, Koshevoy, Gleb
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2016; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Analysis; Applications of Mathematics; Clusters; Complexity; Computation; Computational complexity; Computer Science; Dividing (mathematics); Division; Economics; Graph theory; Linear and Multilinear Algebras; Lower bounds; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Subtraction; Supersymmetry; Transformations; Transformations (mathematics); Trees; United States; Primary 68Q25; Secondary 05E05; Cluster transformation; Arithmetic circuit; Star–mesh transformation; 13F60; Schur function; Subtraction-free; Spanning tree; spanning tree; star-mesh transformation. 2010 Mathematics Subject Classification Primary 68Q25; cluster transformation; arithmetic circuit
• ISSN: 1615-3375; 1615-3383; 1615-3383
• DOI: 10.1007/s10208-014-9231-y

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.