On the Parameterized Intractability of Monadic Second-Order Logic

Details On the Parameterized Intractability of Monadic Second-Order Logic On the Parameterized Intractability of Monadic Second-Order Logic

2012-03-14

arxiv.org/abs/1203.3167v2

LMCS 8 (1:27) 2012

Computer Science

Logic in Computer Science

Scientific paper

10.2168/LMCS-8(1:27)2012

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.

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