## Abstract

We introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive polynomial-space O^{⁎}(3.86^{k})-time (exponential-space O^{⁎}(3.41^{k})-time) deterministic algorithm for k-INTERNAL OUT-BRANCHING, improving upon the previously fastest exponential-space O^{⁎}(5.14^{k})-time algorithm for this problem. (The notation O^{⁎} hides polynomial factors.) We also design polynomial-space O^{⁎}((2e)^{k+o(k)})-time (exponential-space O^{⁎}(4.32^{k})-time) deterministic algorithms for k-COLORFUL OUT-BRANCHING on arc-colored digraphs and k-COLORFUL PERFECT MATCHING on planar edge-colored graphs. In k-COLORFUL OUT-BRANCHING, given an arc-colored digraph D, decide whether D has an out-branching with arcs of at least k colors. k-COLORFUL PERFECT MATCHING is defined similarly. To obtain our polynomial-space algorithms, we show that (n,k,αk)-splitters (α⩾1) and in particular (n,k)-perfect hash families can be enumerated one by one with polynomial delay using polynomial space.

Original language | English |
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Pages (from-to) | 69-85 |

Number of pages | 17 |

Journal | Journal of Computer and System Sciences |

Volume | 95 |

DOIs | |

State | Published - 1 Aug 2018 |

Externally published | Yes |

## Keywords

- Deterministic
- Fixed-parameter tractable
- Kirchoff matrices
- Pfaffians
- Polynomial space