## 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.

Original language | English |
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Title of host publication | 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019) |

Editors | Rolf Niedermeier, Christophe Paul |

Number of pages | 16 |

Volume | 126 |

Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Publication date | 01.03.2019 |

Pages | 11:1--11:16 |

Article number | 11 |

ISBN (Print) | 978-3-95977-100-9 |

DOIs | |

Publication status | Published - 01.03.2019 |

Event | 36th International Symposium on Theoretical Aspects of Computer Science - Berlin, Germany Duration: 13.03.2019 → 16.03.2019 Conference number: 153790 |