## Abstract

We address the carpooling problem as a graph-theoretic problem. If the set of drivers is known in advance, then for any car capacity, the problem is equivalent to the assignment problem in bipartite graphs. Otherwise, when we do not know in advance who will drive their vehicle and who will be a passenger, the problem is NP-hard. We devise and implement quick heuristics for both cases, based on graph algorithms, as well as parallel algorithms based on geometric/algebraic approach. We compare between the algorithms on random graphs, as well as on real, very large, data.

Original language | English |
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Pages (from-to) | 339-347 |

Number of pages | 9 |

Journal | Procedia Computer Science |

Volume | 32 |

DOIs | |

State | Published - 2014 |

Event | 5th International Conference on Ambient Systems, Networks and Technologies, ANT 2014 and 4th International Conference on Sustainable Energy Information Technology, SEIT 2014 - Hasselt, Belgium Duration: 2 Jun 2014 → 5 Jun 2014 |

### Bibliographical note

Funding Information:We have defined the carpooling problem as a graph-theoretic, NP-hard problem. If the drivers of the cars are known in advance then the problem is tractable and is of complexity O(|V|3). We found that the greedy linear algorithm gives close to optimal results in this case. We have also found and implemented quick and efficient incremental solutions for a ‘perturbed’ graph, where some of its edges change. In addition, we found and implemented a fast heuristic algorithm, based on an algebraic approach, which can be parallelized for very large graphs. We suggested and implemented on real data several heuristics for the general intractable problem using linear and almost linear heuristics, and compared between them. Finally, we defined extensions of the carpooling problems where the drivers are known, by allowing the passengers to indicate their priorities, and showed that these extensions are NP-hard. Acknowledgments The research leading to these results has received funding from the European Union Seventh Framework Program (FP7/2007-2013) under grant agreement no 270833.

## Keywords

- Carpooling
- Gradient Projection Algorithm
- Incremental Algorithms
- Linear Programming
- Maximum Weighted Matching
- Scalability
- Star Partition Problem

## ASJC Scopus subject areas

- Computer Science (all)