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On the Descriptive Complexity of Color Coding

Authors Max Bannach, Till Tantau



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Author Details

Max Bannach
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany
Till Tantau
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany

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Max Bannach and Till Tantau. On the Descriptive Complexity of Color Coding. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 11:1-11:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.11

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Fixed parameter tractability
Keywords
  • color coding
  • descriptive complexity
  • fixed-parameter tractability
  • quantifier elimination
  • para-AC^0

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References

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