## Abstract

In the min-cost bipartite perfect matching with delays (MBPMD) problem, requests arrive online at points of a finite metric space. Each request is either positive or negative and has to be matched to a request of opposite polarity. As opposed to traditional online matching problems, the algorithm does not have to serve requests as they arrive, and may choose to match them later at a cost. Our objective is to minimize the sum of the distances between matched pairs of requests (the connection cost) and the sum of the waiting times of the requests (the delay cost). This objective exhibits a natural tradeoff between minimizing the distances and the cost of waiting for better matches. This tradeoff appears in many real-life scenarios, notably, ride-sharing platforms. MBPMD is related to its non-bipartite variant, min-cost perfect matching with delays (MPMD), in which each request can be matched to any other request. MPMD was introduced by Emek et al. (STOC'16), who showed an O(log2 n + log )-competitive randomized algorithm on n-point metric spaces with aspect ratio . Our contribution is threefold. First, we present a new lower bound construction for MPMD and MBPMD.We get a lower bound of q log n log log n on the competitive ratio of any randomized algorithm for MBPMD. For MPMD, we improve the lower bound from (plog n) (shown by Azar et al., SODA'17) to log n log log n , thus, almost matching their upper bound of O(log n). Second, we adapt the algorithm of Emek et al. to the bipartite case, and provide a simplified analysis that improves the competitive ratio to O(log n). The key ingredient of the algorithm is an O(h)- competitive randomized algorithm for MBPMD on weighted trees of height h. Third, we provide an O(h)-competitive deterministic algorithm for MBPMD on weighted trees of height h. This algorithm is obtained by adapting the algorithm for MPMD by Azar et al. to the apparently more complicated bipartite setting.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017 |

Editors | Jose D. P. Rolim, Klaus Jansen, David P. Williamson, Santosh S. Vempala |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770446 |

DOIs | |

State | Published - 1 Aug 2017 |

Event | 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, United States Duration: 16 Aug 2017 → 18 Aug 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 81 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 |
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Country/Territory | United States |

City | Berkeley |

Period | 16/08/17 → 18/08/17 |

### Funding

Funders | Funder number |
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Israel Science Foundation | 1506/16 |

Israeli Centers for Research Excellence | 4/11 |

## Keywords

- Bipartite matching
- Competitive analysis
- Online algorithms with delayed service