### Abstract

Color Coding is an algorithmic technique for deciding efficiently if a given input graph contains a path of a given length (or another small subgraph of constant tree-width). Applications of the method in computational biology motivate the study of similar algorithms for counting the number of copies of a given subgraph. While it is unlikely that exact counting of this type can be performed efficiently, as the problem is #W[1]-complete even for paths, approximate counting is possible, and leads to the investigation of an intriguing variant of families of perfect hash functions. A family of functions from [n] to [k] is an (ε,k)-balanced family of hash functions, if there exists a positive T so that for every K ⊂ [n] of size |K| = k, the number of functions in the family that are one-to-one on K is between (1 - ε)T and (1 + ε)T. The family is perfectly k-balanced if it is (0,k)-balanced. We show that every such perfectly k-balanced family is of size at least c(k)n ^{[k/2]}, and that for every ε > 1/poly(k) there are explicit constructions of (ε,k)-balanced families of hash functions from [n] to [k] of size e^{(1+o(1))k} logn. This is tight up to the o(1)-term in the exponent, and supplies deterministic polynomial time algorithms for approximately counting the number of paths or cycles of a specified length k (or copies of any graph H with k vertices and bounded tree-width) in a given input graph of size n, up to relative error ε, for all k ≤ O(logn).

Original language | English (US) |
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Title of host publication | Parameterized and Exact Computation - 4th International Workshop, IWPEC 2009, Revised Selected Papers |

Pages | 1-16 |

Number of pages | 16 |

DOIs | |

State | Published - Dec 24 2009 |

Externally published | Yes |

Event | 4th International Workshop on Parameterized and Exact Computation, IWPEC 2009 - Copenhagen, Denmark Duration: Sep 10 2009 → Sep 11 2009 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5917 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th International Workshop on Parameterized and Exact Computation, IWPEC 2009 |
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Country | Denmark |

City | Copenhagen |

Period | 9/10/09 → 9/11/09 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Keywords

- Approximate counting of subgraphs
- Color-coding
- Derandomization
- Expanders
- K-wise independence
- Perfect hashing

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## Cite this

*Parameterized and Exact Computation - 4th International Workshop, IWPEC 2009, Revised Selected Papers*(pp. 1-16). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5917 LNCS). https://doi.org/10.1007/978-3-642-11269-0_1