On the Parameterized Intractability of Monadic Second-Order Logic

TL;DR

This paper investigates the limits of fixed-parameter tractability for monadic second-order logic model-checking on graph classes with unbounded tree-width, establishing complexity boundaries based on tree-width growth.

Contribution

It demonstrates that extending Courcelle's theorem to classes with unbounded tree-width is unlikely unless major complexity-theoretic conjectures fail.

Findings

MSO_2-model checking is not fixed-parameter tractable if tree-width exceeds log^{84} n unless SAT is in sub-exponential time.

For classes with non-poly-logarithmic unbounded tree-width, MSO_2-model checking is not fpt unless the polynomial hierarchy collapses.

The results set near-optimal boundaries for the tractability of MSO_2 logic on complex graph classes.

Abstract

One of Courcelle's celebrated results states that if C is a class of graphs of bounded tree-width, then model-checking for monadic second order logic (MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized algorithms, where the parameter is the tree-width plus the size of the formula. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded tree-width. In this paper we show that in terms of tree-width, the theorem cannot be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions and is such that the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is not fpt unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt unless all problems in the polynomial-time hierarchy can be solved in sub-exponential time.