Goyal, Prachi and Misra, Neeldhara and Panolan, Fahad and Zehavi, Meirav
(2015)
*DETERMINISTIC ALGORITHMS FOR MATCHING AND PACKING PROBLEMS BASED ON REPRESENTATIVE SETS.*
In: SIAM JOURNAL ON DISCRETE MATHEMATICS, 29
(4).
pp. 1815-1836.

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## 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 subset of U-1 x ... x 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 subset of 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) .vertical bar F vertical bar. n log(2)n . logW) 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).vertical bar F vertical bar.n log(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).vertical bar F vertical bar . n log(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) logW) 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.

Item Type: | Journal Article |
---|---|

Publication: | SIAM JOURNAL ON DISCRETE MATHEMATICS |

Publisher: | SIAM PUBLICATIONS |

Additional Information: | Copy right for this article belongs to the SIAM PUBLICATIONS, 3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA |

Keywords: | r-dimensional matching; set packing; 3D-matching; fixed-parameter algorithms; representative sets; iterative expansion |

Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |

Date Deposited: | 20 Jan 2016 04:55 |

Last Modified: | 20 Jan 2016 04:55 |

URI: | http://eprints.iisc.ac.in/id/eprint/53160 |

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