Output-sensitive algorithm for generating the flats of a matroid

TL;DR

This paper introduces an output-sensitive algorithm for generating all flats of a finite matroid, with applications to efficiently computing zonotopes in H-representation, improving computational methods in matroid theory.

Contribution

The paper presents a novel output-sensitive algorithm for generating matroid flats and a specialized efficient algorithm for vectorial matroids, enhancing computational tools in matroid theory.

Findings

Algorithm has complexity O(N^2 M S_P) for general matroids.

Specialized algorithm for vectorial matroids runs in O(N^2 M d^2) time.

Potential application in efficient zonotope computation from Minkowski sums.

Abstract

We present an output-sensitive algorithm for generating the whole set of flats of a finite matroid. Given a procedure, P, that decides in S_P time steps if a set is independent, the time complexity of the algorithm is O(N^2 M S_P), where N and M are the input and output size, respectively. In the case of vectorial matroids, a specific algorithm is reported whose time complexity is equal to O(N^2 M d^2), d being the rank of the matroid. In some cases this algorithm can provide an efficient method for computing zonotopes in $H$-representation, given their representation in terms of Minkowski sum of known segments.