# Matroids Hitting Sets and Unsupervised Dependency Grammar Induction

## TL;DR

This paper introduces a new graph problem related to minimal edge sets that encompass all spanning trees of certain subgraphs, inspired by unsupervised grammar induction, and offers theoretical analysis and algorithms.

## Contribution

It formulates a novel graph problem motivated by unsupervised dependency grammar induction and provides reductions, complexity analysis, and an approximation algorithm.

## Key findings

- Problem is computationally complex.
- Reduction to known graph problems established.
- Approximation algorithm proposed.

## Abstract

This paper formulates a novel problem on graphs: find the minimal subset of edges in a fully connected graph, such that the resulting graph contains all spanning trees for a set of specifed sub-graphs. This formulation is motivated by an un-supervised grammar induction problem from computational linguistics. We present a reduction to some known problems and algorithms from graph theory, provide computational complexity results, and describe an approximation algorithm.