## Abstract

The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph G= (V, A) , a capacity function c: V→ N, and a weight function w: A→ R^{+}, a carpool matching is a subset of arcs, M⊆ A, such that every v∈ V satisfies: (1) dMin(v)·dMout(v)=0, (2) dMin(v)≤c(v), and (3) dMout(v)≤1. A vertex v for which dMout(v)=1 is a passenger, and a vertex for which dMout(v)=0 is a driver who has dMin(v) passengers. In the Maximum Carpool Matching problem the goal is to find a carpool matching M of maximum total weight. The problem arises when designing an online carpool service, such as Zimride (Zimride by enterprise. https://zimride.com/), which tries to connect between users based on a similarity function. The problem is known to be NP-hard, even in the unweighted and uncapacitated case. The Maximum Group Carpool Matching problem, is an extension of the Maximum Carpool Matching where each vertex represents an unsplittable group of passengers. Formally, each vertex u∈ V has a size s(u) ∈ N, and the constraint dMin(v)≤c(v) is replaced with ∑ _{u}_{:}_{(}_{u}_{,}_{v}_{)}_{∈}_{M}s(u) ≤ c(v). We show that Maximum Carpool Matching can be formulated as an unconstrained submodular maximization problem, thus it admits a 12-approximation algorithm. We show that the same formulation does not work for Maximum Group Carpool Matching, nevertheless, we present a local search (12-ε)-approximation algorithm for Maximum Group Carpool Matching. For the unweighted variant of both problems when the maximum possible capacity, c_{max}, is bounded by a constant, we provide a local search (12+12cmax-ε)-approximation algorithm. We also provide an APX-hardness result, even if the maximum degree and c_{max} are at most 3.

Original language | English |
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Pages (from-to) | 3165-3182 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 82 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2020 |

### Bibliographical note

Publisher Copyright:© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

### Funding

A preliminary version was presented at the 25th Annual European Symposium on Algorithms, 2017. D. Rawitz: Supported in part by the Israel Science Foundation (Grant No. 497/14).

Funders | Funder number |
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Israel Science Foundation | 497/14 |

## Keywords

- Approximation algorithms
- Local search
- Star packing
- Submodular maximization