Decomposition width - a new width parameter for matroids

TL;DR

This paper introduces decomposition width, a new parameter for matroids, enabling linear-time algorithms for monadic second order logic properties on matroids with bounded decomposition width, extending previous results to non-representable matroids.

Contribution

The paper defines decomposition width for matroids and demonstrates its utility in enabling efficient algorithms for a broad class of matroid properties, extending prior work.

Findings

Linear-time algorithms for MSO properties on matroids with bounded decomposition width

Decomposition width can be computed in polynomial time for certain matroids

Extends algorithmic results to non-representable matroids

Abstract

We introduce a new width parameter for matroids called decomposition width and prove that every matroid property expressible in the monadic second order logic can be computed in linear time for matroids with bounded decomposition width if their decomposition is given. Since decompositions of small width for our new notion can be computed in polynomial time for matroids of bounded branch-width represented over finite fields, our results include recent algorithmic results of Hlineny [J. Combin. Theory Ser. B 96 (2006), 325-351] in this area and extend his results to matroids not necessarily representable over finite fields.