# Optimizing tree decompositions in MSO

## TL;DR

This paper demonstrates that optimal tree decompositions can be obtained via monadic second-order logic (MSO) transductions in fixed-parameter linear time, extending classic algorithms with a logical framework.

## Contribution

It proves that optimal tree decompositions are MSO-definable and implementable in linear fixed-parameter time, unifying logical and algorithmic approaches.

## Key findings

- MSO transductions can produce optimal tree decompositions.
- The approach generalizes Bodlaender and Kloks' linear fixed-parameter algorithm.
- Optimal decompositions are definable and computable within MSO logic.

## Abstract

The classic algorithm of Bodlaender and Kloks [J. Algorithms, 1996] solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in mso in the following sense: for every positive integer k, there is an mso transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results [LICS 2016], this implies that for every k there exists an mso transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width. We also show that mso transductions can be implemented in linear fixed-parameter time, which enables us to derive the algorithmic result of Bodlaender and Kloks as a corollary of our main result.