Expanding the expressive power of Monadic Second-Order logic on restricted graph classes

TL;DR

This paper develops fixed-parameter tractable algorithms for complex Monadic Second-Order logic problems on graphs with bounded vertex cover, extending the scope of efficient algorithms to problems previously considered hard.

Contribution

It introduces two new meta-theorems enabling FPT algorithms for cardMSO1 and MSO partitioning problems on graphs with bounded vertex cover, using ILP techniques.

Findings

FPT algorithms for cardMSO1 on bounded vertex cover graphs

FPT algorithms for Rao's MSO partitioning problems

Extension of results from vertex cover to neighborhood diversity

Abstract

We combine integer linear programming and recent advances in Monadic Second-Order model checking to obtain two new algorithmic meta-theorems for graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of the well-known Monadic Second-Order logic by the addition of cardinality constraints, can be solved in FPT time parameterized by vertex cover. The second meta-theorem shows that the MSO partitioning problems introduced by Rao can also be solved in FPT time with the same parameter. The significance of our contribution stems from the fact that these formalisms can describe problems which are W[1]-hard and even NP-hard on graphs of bounded tree-width. Additionally, our algorithms have only an elementary dependence on the parameter and formula. We also show that both results are easily extended from vertex cover to neighborhood diversity.