Recognizing Series-Parallel Matrices in Linear Time

TL;DR

This paper presents a linear-time algorithm to recognize series-parallel matrices, which are linked to graphic matroids of series-parallel graphs, and provides an implementation with computational results.

Contribution

The paper introduces a novel linear-time algorithm for recognizing series-parallel matrices and offers an efficient implementation with experimental validation.

Findings

Algorithm runs in expected O(m + n + k) time

Successfully identifies series-parallel matrices in practice

Provides minimal non-series-parallel submatrix when recognition fails

Abstract

A series-parallel matrix is a binary matrix that can be obtained from an empty matrix by successively adjoining rows or columns that are parallel to an existing row/column or have at most one 1-entry. Equivalently, series-parallel matrices are representation matrices of graphic matroids of series-parallel graphs, which can be recognized in linear time. We propose an algorithm that, for an m-by-n matrix A with k nonzeros, determines in expected $\mathcal{O}(m + n + k)$ time whether A is series-parallel, or returns a minimal non-series-parallel submatrix of A. We complement the developed algorithm by an efficient implementation and report about computational results.