## Abstract

In the classic problem of fair cake-cutting, a single interval (“cake”) has to be divided among n agents with different value measures, giving each agent a single sub-interval with a value of at least 1∕n of the total. This paper studies a generalization in which the cake is made of m disjoint intervals, and each agent should get at most k sub-intervals. The paper presents a polynomial-time algorithm that guarantees to each agent at least min(1∕n,k∕(m+n−1)) of the total value, and shows that this is the largest fraction that can be guaranteed. The algorithm simultaneously guarantees to each agent at least 1∕n of the value of his or her k most valuable islands. The main technical tool is envy-free matching in a bipartite graph. Some of the results remain valid even with additional fairness constraints such as envy-freeness. Besides the natural application of the algorithm to simultaneous division of multiple land-estates, the paper shows an application to a geometric problem — fair division of a two-dimensional land estate shaped as a rectilinear polygon, where each agent should receive a rectangular piece.

Original language | English |
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Pages (from-to) | 15-35 |

Number of pages | 21 |

Journal | Discrete Applied Mathematics |

Volume | 291 |

DOIs | |

State | Published - 11 Mar 2021 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier B.V.

### Funding

I am grateful to Elad Aigner-Horev, Chris Culter, 10 10 Zur Luria, Yuval Filmus, Gregory Nisbet and lulu for their helpful ideas, and three anonymous reviewers of Discrete Applied Mathematics for their many constructive comments. The research was partly funded by the Doctoral Fellowships of Excellence Program at Bar-Ilan University , the Mordecai and Monique Katz Graduate Fellowship Program , and the Israel Science Foundation grants 1083/13 and 712/20 .

Funders | Funder number |
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Bar-Ilan University | |

Israel Science Foundation | 1083/13, 712/20 |

## Keywords

- Cutting
- Fair division
- Matching
- Rectilinear polygon