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We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudopolynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time.We also consider the class of k-ary instances where k is a constant, i.e., each agent has at most k different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations.We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.
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Equitable Allocations of Indivisible Goods
In fair division,equitabilitydictates that each partic-ipant receives the same level of utility. In this work,we study equitable allocations of indivisible goodsamong agents with additive valuations. While priorwork has studied (approximate) equitability in iso-lation, we consider equitability in conjunction withother well-studied notions of fairness and economicefficiency. We show that the Leximin algorithm pro-duces an allocation that satisfies equitability up toany good and Pareto optimality. We also give anovel algorithm that guarantees Pareto optimalityand equitability up to one good in pseudopolyno-mial time. Our experiments on real-world prefer-ence data reveal that approximate envy-freeness, ap-proximate equitability, and Pareto optimality canoften be achieved simultaneously.
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- PAR ID:
- 10172920
- Date Published:
- Journal Name:
- IJCAI
- ISSN:
- 1045-0823
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.24963/ijcai.2019/40
