Amalgam width of matroids

TL;DR

This paper introduces the amalgam width parameter for matroids, enabling linear-time decision procedures for monadic second order properties and polynomial-time computation of the Tutte polynomial for matroids with bounded amalgam width.

Contribution

It defines a new matroid width parameter called amalgam-width, which is linearly related to branch-width and allows efficient algorithms for certain properties.

Findings

Decidable monadic second order properties in linear time for bounded amalgam-width.

Polynomial-time computation of the Tutte polynomial for matroids with bounded amalgam width.

Amalgam-width is linearly related to branch-width on finitely representable matroids.

Abstract

We introduce a new matroid width parameter based on the operation of matroid amalgamation, which we call amalgam-width. The parameter is linearly related to branch-width on finitely representable matroids (which is not possible for branch-width). In particular, any property expressible in the monadic second order logic can be decided in linear time for matroids with bounded amalgam-width. We also prove that the Tutte polynomial can be computed in polynomial time for matroids with bounded amalgam width.