Lower bounds on the complexity of MSO1MSO1 model-checking

• Efficient coloured MSO1 properties on monotone graph classes need small tree-width.
• This extends recent Kreutzer–Tazari to a different class of properties.
• There is likely no monotone and algorithmically useful digraph width measures.

Kreutzer and Tazari proved in 2010 that MSO2MSO2 model-checking is not polynomial (XP) w.r.t. the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. We prove that MSO1MSO1 model-checking with a fixed set of vertex labels — i.e., without edge-set quantification — is not solvable even in quasi-polynomial time for fixed MSO1MSO1-formulas in such graph classes. Both the lower bounds hold modulo a certain complexity-theoretic assumption, namely, the Exponential-Time Hypothesis (ETH) in the former case and the non-uniform ETH in the latter case. In comparison to Kreutzer and Tazari, we show a different set of problems to be intractable, and our stronger complexity assumption of non-uniform ETH slightly weakens assumptions on the graph class and greatly simplifies important lengthy parts of the former proof. Our result also has an interesting consequence in the realm of digraph width measures.