On the Descriptive Complexity of Color Coding

Max Bannach; Till Tantau

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arXiv · 1901.03364

On the Descriptive Complexity of Color Coding

Max Bannach·Till Tantau

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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.

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Max Bannach, Till Tantau. 2019-01-10. On the Descriptive Complexity of Color Coding. https://arxiv.org/abs/1901.03364

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