## Abstract

In this work, we study the well-known r-DIMENSIONAL k-MATCHING ((r, k)-DM), and r-SET k-PACKING ((r, k)-SP) problems. Given a universe U := U_{1} ⋯ U_{r} and an r-uniform family F ⊆ U_{1} × ⋯ × U_{r}, the (r, k)-DM problem asks if F admits a collection of k mutually disjoint sets. Given a universe U and an r-uniform family F ⊆ 2^{U}, the (r,k)SP problem asks if F admits a collection of k mutually disjoint sets. We employ techniques based on dynamic programming and representative families. This leads to a deterministic algorithm with running time O(2.851^{(r-1)k}. |F|. n log^{2} n-log W) for the weighted version of (r, k)-DM, where W is the maximum weight in the input, and a deterministic algorithm with running time O(2.851(r-0.5501)k · |F| · nlog^{2} n · logW) for the weighted version of (r, k)SP. Thus, we significantly improve the previous best known deterministic running times for (r, k)-DM and (r, k)SP and the previous best known running times for their weighted versions. We rely on structural properties of (r, k)-DM and (r, k)SP to develop algorithms that are faster than those that can be obtained by a standard use of representative sets. Incorporating the principles of iterative expansion, we obtain a better algorithm for (3,k)-DM, running in time O(2.004^{3k} · |F| · nlog^{2} n). We believe that this algorithm demonstrates an interesting application of representative families in conjunction with more traditional techniques. Furthermore, we present kernels of size O(e^{r}r(k - 1)^{r} log W) for the weighted versions of (r, k)-DM and (r, k)-SP, improving the previous best known kernels of size O(r!r(k-1)^{r} logW) for these problems.

Original language | English |
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Pages (from-to) | 1815-1836 |

Number of pages | 22 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2015 |

Externally published | Yes |

## Keywords

- 3D-matching
- Fixed-parameter algorithms
- Iterative expansion
- R-dimensional matching
- Representative sets
- Set packing

## ASJC Scopus subject areas

- Mathematics (all)