Deciding first order logic properties of matroids

TL;DR

This paper extends the concept of locally bounded tree-width from graphs to matroids, demonstrating fixed parameter tractability for first order logic properties in regular matroids with locally bounded branch-width.

Contribution

It introduces the notion of locally bounded branch-width for matroids and proves fixed parameter algorithms for first order logic properties in this class.

Findings

Deciding the existence of short circuits containing specific elements is fixed parameter tractable for regular matroids.

First order logic properties can be decided efficiently in classes of regular matroids with locally bounded branch-width.

Abstract

Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.