Approximation and complexity of the optimization and existence problems for maximin share, proportional share, and minimax share allocation of indivisible goods

This paper is concerned with various types of allocation problems in fair division of indivisible goods, aiming at maximin share, proportional share, and minimax share allocations. However, such allocations do not always exist, not even in very simple settings with two or three agents. A natural que...

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Bibliographic Details
Published in:Autonomous agents and multi-agent systems Vol. 32; no. 6; pp. 741 - 778
Main Authors: Heinen, Tobias, Nguyen, Nhan-Tam, Nguyen, Trung Thanh, Rothe, Jörg
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2018
Springer Nature B.V
Subjects:
Allocations
Approximation
Artificial Intelligence
Computer Science
Computer Systems Organization and Communication Networks
Economic models
Mathematical analysis
Minimax technique
Reagents
Software Engineering/Programming and Operating Systems
User Interfaces and Human Computer Interaction
Utilities
Utility functions
Computational social choice
Approximation
Maximin share
Resource allocation
Indivisible goods
Minimax share
Fair division
Proportional share
Complexity
ISSN:1387-2532, 1573-7454
Online Access:Get full text
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Summary:This paper is concerned with various types of allocation problems in fair division of indivisible goods, aiming at maximin share, proportional share, and minimax share allocations. However, such allocations do not always exist, not even in very simple settings with two or three agents. A natural question is to ask, given a problem instance, what is the largest value c for which there is an allocation such that every agent has utility of at least c times her fair share. We first prove that the decision problem of checking if there exists a minimax share allocation for a given problem instance is NP -hard when the agents’ utility functions are additive. We then show that, for each of the three fairness notions, one can approximate c by a polynomial-time approximation scheme, assuming that the number of agents is fixed. Next, we investigate the restricted cases when utility functions have values in { 0 , 1 } only or are defined based on scoring vectors (Borda and lexicographic vectors), and we obtain several tractability results for these cases. Interestingly, we show that maximin share allocations can always be found efficiently with Borda utilities, which cannot be guaranteed for general additive utilities. In the nonadditive setting, we show that there exists a problem instance for which there is no c -maximin share allocation, for any constant c . We explore a class of symmetric submodular utilities for which there exists a tight 1 2 -maximin share allocation, and show how it can be approximated to within a factor of  1 4 .
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ISSN:1387-2532
1573-7454
DOI:10.1007/s10458-018-9393-0