Problem
Research questions and friction points this paper is trying to address.
Courcelle's theorem
fine-grained complexity
treewidth
monadic second-order logic
Exponential Time Hypothesis
Fine-Grained Bounds for Courcelle's Theorem
🤖 AI Summary
This work addresses the substantial gap between known upper and lower bounds for monadic second-order (MSO) model checking on graphs of bounded treewidth, a problem governed by Courcelle’s Theorem. While prior results offered coarse time complexity estimates that largely ignored the internal quantifier structure of MSO formulas, this paper presents the first nearly tight upper bound that explicitly depends on both the treewidth and the number of first- and second-order variables within each quantifier alternation block of the formula. By integrating parameterized complexity analysis, fine-grained reductions, and the Exponential Time Hypothesis (ETH), the study significantly narrows the theoretical gap between upper and lower bounds, thereby providing the most refined characterization to date of the time complexity inherent in Courcelle’s Theorem.
📝 Abstract
Courcelle's theorem states that there exists an algorithm that takes as input a graph $G$ of treewidth at most $t$ and a MSO formula $φ$, and determines whether $G$ satisfies $φ$ in time $f(φ,t) \cdot n$. It is folklore that the the function $f$ contains a tower of exponentials whose height depends as a linear function of the number of quantifier alternations of the input formula $φ$. A classic reduction of Frick and Grohe shows that, assuming the Exponential Time Hypothesis (ETH), the linear growth of the height of the tower is unavoidable. Nevertheless, there is still a huge gap between existing upper and lower bounds -- after all, there is quite a difference between a single exponential and a double exponential running time. In addition, this only gives us a very coarse understanding in the time complexity of Courcelle's theorem. In this paper, we prove a fine-grained version of Courcelle's theorem with nearly ETH-tight dependence on the treewidth parameter $t$ and the quantifier structure of $φ$ (specifically, the number of first order and second order variables in each quantifier alternation block).
Innovation
Methods, ideas, or system contributions that make the work stand out.
fine-grained complexity
Courcelle's theorem
treewidth
monadic second-order logic
Exponential Time Hypothesis
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