research-article

High-Multiplicity Fair Allocation: Lenstra Empowered by N-fold Integer Programming

Authors:

Robert Bredereck,

Andrzej Kaczmarczyk,

Rolf NiedermeierAuthors Info & Claims

EC '19: Proceedings of the 2019 ACM Conference on Economics and Computation

Pages 505 - 523

https://doi.org/10.1145/3328526.3329649

Published: 17 June 2019 Publication History

Abstract

We study the (parameterized) computational complexity of problems in the context of fair allocations of indivisible goods. More specifically, we show fixed-parameter tractability results for a broad set of problems concerned with envy-free, Pareto-efficient allocations of items (with agent-specific utility functions) to agents. In principle, this implies efficient exact algorithms for these in general computationally intractable problems whenever we face instances with few agents and low maximum (absolute) utility values. This holds true also in high-multiplicity settings where we may have high numbers of identical items. On the technical side, our approach provides algorithmic meta-theorems covering a rich set of fair allocation problems in the additive preferences model. To achieve this, our main technical contribution is to make an elaborate use of tools from integer linear programming. More specifically, we exploit results originally going back to a famous theorem of Lenstra [Math. Oper. Res. 1983] concerning (the fixed-parameter tractability of) Integer Linear Programs (ILPs) with bounded dimension (that is, the dimension shall be considered as a (small) parameter) and the more recent framework of (combinatorial) N-fold ILPs. We reveal and exploit a fruitful interaction between these two cornerstones in the theory of integer linear programming, which may be of independent interest in applications going beyond fair allocations.

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Cited By

Briański MKoutecký MKrál’ DPekárková KSchröder F(2024)Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programmingMathematical Programming10.1007/s10107-023-02048-x208:1-2(497-531)Online publication date: 20-Jan-2024
https://doi.org/10.1007/s10107-023-02048-x
Eiben EGanian RHamm TOrdyniak S(2022)Parameterized Complexity of Envy-Free Resource Allocation in Social NetworksArtificial Intelligence10.1016/j.artint.2022.103826(103826)Online publication date: Nov-2022
https://doi.org/10.1016/j.artint.2022.103826
Nguyen TRothe J(2022)Fair and Efficient Allocation with Few Agent Types, Few Item Types, or Small Value LevelsArtificial Intelligence10.1016/j.artint.2022.103820(103820)Online publication date: Nov-2022
https://doi.org/10.1016/j.artint.2022.103820
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Index Terms

High-Multiplicity Fair Allocation: Lenstra Empowered by N-fold Integer Programming
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Discrete optimization
2. Theory of computation
  1. Theory and algorithms for application domains
    1. Algorithmic game theory and mechanism design
      1. Algorithmic game theory
      2. Market equilibria

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Published In

cover image ACM Conferences

EC '19: Proceedings of the 2019 ACM Conference on Economics and Computation

June 2019

947 pages

ISBN:9781450367929

DOI:10.1145/3328526

General Chair:
Anna Karlin
University of Washington, USA
,
Program Chairs:
Nicole Immorlica
Microsoft Research, USA
,
Ramesh Johari
Stanford University, USA

Copyright © 2019 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Published: 17 June 2019

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Deutsche Forschungsgemeinschaft
Czech Science Foundation

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EC '19

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SIGecom

EC '19: ACM Conference on Economics and Computation

June 24 - 28, 2019

AZ, Phoenix, USA

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EC '19 Paper Acceptance Rate 106 of 382 submissions, 28%;

Overall Acceptance Rate 664 of 2,389 submissions, 28%

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The 25th ACM Conference on Economics and Computation

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Cited By

Briański MKoutecký MKrál’ DPekárková KSchröder F(2024)Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programmingMathematical Programming10.1007/s10107-023-02048-x208:1-2(497-531)Online publication date: 20-Jan-2024
https://doi.org/10.1007/s10107-023-02048-x
Eiben EGanian RHamm TOrdyniak S(2022)Parameterized Complexity of Envy-Free Resource Allocation in Social NetworksArtificial Intelligence10.1016/j.artint.2022.103826(103826)Online publication date: Nov-2022
https://doi.org/10.1016/j.artint.2022.103826
Nguyen TRothe J(2022)Fair and Efficient Allocation with Few Agent Types, Few Item Types, or Small Value LevelsArtificial Intelligence10.1016/j.artint.2022.103820(103820)Online publication date: Nov-2022
https://doi.org/10.1016/j.artint.2022.103820
Mishra SPadala MGujar S(2022)EEF1-NN: Efficient and EF1 Allocations Through Neural NetworksPRICAI 2022: Trends in Artificial Intelligence10.1007/978-3-031-20865-2_29(388-401)Online publication date: 4-Nov-2022
https://doi.org/10.1007/978-3-031-20865-2_29
Misra NNayak D(2022)On Fair Division with Binary Valuations Respecting Social NetworksAlgorithms and Discrete Applied Mathematics10.1007/978-3-030-95018-7_21(265-278)Online publication date: 24-Jan-2022
https://doi.org/10.1007/978-3-030-95018-7_21
Bredereck RFigiel AKaczmarczyk AKnop DNiedermeier RDignum FLomuscio AEndriss UNowé A(2021)High-Multiplicity Fair Allocation Made More PracticalProceedings of the 20th International Conference on Autonomous Agents and MultiAgent Systems10.5555/3463952.3463988(260-268)Online publication date: 3-May-2021
https://dl.acm.org/doi/10.5555/3463952.3463988
Bredereck RFigiel AKaczmarczyk AKnop DNiedermeier RDignum FLomuscio AEndriss UNowé A(2021)High-Multiplicity Fair Allocation Made More PracticalProceedings of the 20th International Conference on Autonomous Agents and MultiAgent Systems10.5555/3461017.3461052(260-268)Online publication date: 3-May-2021
Bulmer JFritter MGao YHui B(2020)FASTT: Team Formation Using Fair DivisionAdvances in Artificial Intelligence10.1007/978-3-030-47358-7_9(92-104)Online publication date: 6-May-2020
https://doi.org/10.1007/978-3-030-47358-7_9

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