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STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 53/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Lower Bounds for Monadic Second-Order Logic We would like to show. If a class C of graphs has unbounded tree-width then MC(MSO 2 ,C) is not fixed-parameter tractable. Sadly, in this generality this is not true. Theorem. (Makowsky, Mariño 04) There are classes C of graphs of unbounded tree-width on which MC(MSO 2 ,C) is tractable. But something similar is true. Unbounded Tree-Width. We first need to classify the unboundedness of tree-width.
STEPHAN KREUTZER COMPLEXITY OF MODEL-CHECKING PROBLEMS 54/81 INTRODUCTION COMPLEXITY UPPER BOUNDS COMPOSITION LOCALITY LOCALISATION GRIDS GRID-LIKE MINORS LABELLED WEBS Classes of Unbounded Tree-Width Definition. Let f : N → N be a non-decreasing function. A class C of graphs has f -bounded tree-width if tw(G) ≤ f(|G|) for all G ∈ C. Examples. • In Courcelle’s theorem, f(n) := c is constant. • f(n) := n is the maximal function that makes sense here. • We will look at f(n) := log c n for a small constant c > 0. Theorem by Makowsky, Mariño. There are classes C of graphs of logarithmic tree-width on which MC(MSO 2 ,C) is tractable. What we would like to show. If the tree-width of C is not bounded by log c n, for small constant c, then MC(MSO,C) is not FPT.
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