[Paper Review] Subtraction-free complexity, cluster transformations, and spanning trees

Sergey Fomin, D. Grigoriev+1 more|HAL (Le Centre pour la Communication Scientifique Directe)|Jul 31, 2013

Complexity and Algorithms in Graphs25 references4 citations

TL;DR

This paper introduces subtraction-free algorithms using cluster transformations to compute Schur functions, their skew, double, and supersymmetric variants, and generating functions of spanning trees in polynomial time. It demonstrates that division can be exponentially powerful in subtraction-free complexity, showing an exponential gap between subtraction-free and ordinary arithmetic circuit complexity for certain rational functions.

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.

Motivation & Objective

• To investigate how restricting arithmetic operations to only addition, multiplication, and division (excluding subtraction) affects computational complexity.
• To develop efficient subtraction-free algorithms for computing Schur functions and their generalizations using cluster transformations.
• To design polynomial-time subtraction-free algorithms for generating functions of both directed and undirected spanning trees.
• To demonstrate that division can be exponentially powerful in the absence of subtraction by showing exponential gaps between subtraction-free and ordinary circuit complexity.
• To construct explicit examples where subtraction-free complexity grows exponentially despite linear ordinary complexity.

Proposed method

• Utilizes cluster transformations—rational maps central to cluster algebra theory—as the core mechanism for designing subtraction-free algorithms.
• Applies the star-mesh transformation to reduce the complexity of computing generating functions for spanning trees in graphs.
• Employs Pólya's theorem on positive polynomials to ensure that rational functions with nonnegative coefficients can be expressed as ratios of polynomials with nonnegative coefficients.
• Uses degree bounds on numerator and denominator polynomials to derive lower bounds on subtraction-free circuit complexity via Lemma 8.1.
• Constructs specific rational functions, such as $ G_n(x) = (x-1)^2 + 2^{-2^{n+1}} $, to demonstrate exponential lower bounds on subtraction-free complexity.
• Applies the Lazard-Mora-Philippon trick to create a homogeneous polynomial $ H_n $ with finite but super-exponential subtraction-free complexity, while maintaining linear ordinary complexity.

Experimental results

Research questions

• RQ1 Can Schur functions and their generalizations be computed efficiently in the subtraction-free model using cluster transformations?
• RQ2 What is the computational complexity of generating functions for spanning trees under subtraction-free arithmetic circuits?
• RQ3 How does the absence of subtraction affect the power of division in arithmetic circuits?
• RQ4 Can there be an exponential gap between ordinary and subtraction-free circuit complexity for rational functions?
• RQ5 Are there explicit rational functions where subtraction-free complexity grows exponentially despite linear ordinary complexity?

Key findings

• The paper presents subtraction-free polynomial-time algorithms for computing Schur functions, skew Schur functions, double Schur functions, and supersymmetric Schur functions using cluster transformations.
• It develops subtraction-free algorithms for generating functions of both directed and undirected spanning trees, achieving polynomial complexity.
• The subtraction-free complexity of the univariate polynomial $ G_n(x) = (x-1)^2 + 2^{-2^{n+1}} $ exceeds $ 2^n $, despite its ordinary complexity being $ O(1) $ with arbitrary constants or $ O(n) $ with only 1 as input.
• For the homogeneous polynomial $ H_n(t,x_1,\dots,x_n) $, the subtraction-free complexity exceeds $ 2^{n-2} $, while its ordinary complexity is linear in $ n $.
• The study confirms that in the absence of subtraction, division can be exponentially powerful, as shown by the exponential gap between subtraction-free and semiring (addition/multiplication-only) complexity.
• The results extend to the tropical semifield, implying that the complexity of the minimum cost arborescence problem drops from exponential to polynomial when subtraction is allowed.