Abstract
Color coding is an algorithmic technique used in parameterized complexity theory to detect “small” structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing—purely in terms of the syntactic structure of describing formulas—when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in FPT just because of the way they are commonly described using logical formulas.
Highlights
Descriptive complexity provides a powerful link between logic and complexity theory: We use a logical formula to describe a problem and can infer the computational complexity of the problem just from the syntactic structure of the formula
Our third contribution is a new result in the same vein as the already repeatedly mentioned result of Chen et al [4], stated as Fact 1 in our paper: Our new Theorem 1 states that a parameterized problem can be described slicewise by a familyk∈N of arithmetic first-order formulas that all use only a bounded number of variables if and only if the problem lies in para-AC0↑ —a class that has been encountered repeatedly in the literature [5,7,8,9], but for which no characterization was known
The crucial difference is, that we identify syntactic properties of logical formulas that imply that the color coding technique can be applied
Summary
Descriptive complexity provides a powerful link between logic and complexity theory: We use a logical formula to describe a problem and can infer the computational complexity of the problem just from the syntactic structure of the formula. Our third contribution is a new result in the same vein as the already repeatedly mentioned result of Chen et al [4], stated as Fact 1 in our paper: Our new Theorem 1 states that a parameterized problem can be described slicewise by a family (φk )k∈N of arithmetic first-order formulas that all use only a bounded number of variables if and only if the problem lies in para-AC0↑ —a class that has been encountered repeatedly in the literature [5,7,8,9], but for which no characterization was known. By Theorem 3 we see that the above formulas are equivalent to a family of formulas with a bounded number of variables and by Theorem 1 we see that pk - LONG - PATH ∈ para-AC0↑ ⊆ FPT These ideas generalize and we give a purely syntactic proof of the seminal result from the original color coding paper [3] that the embedding problem for graphs of bounded tree width lies in FPT.
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