Approximation Algorithms for Computing Maximin Share Allocations (1503.00941v3)

Abstract: We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of $n$ agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, so we resort to approximation algorithms. Our main result is a $2/3$-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang, which also produces a $2/3$-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of their algorithm. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in relevant works. Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of $3$ agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of $2$ agents). We provide a $7/8$-approximation algorithm, improving the previously known result of $3/4$. The second case is when all item values belong to ${0, 1, 2}$, extending the ${0, 1}$ setting studied in Bouveret and Lema^itre. We obtain an exact algorithm for any number of agents in this case.

• The paper presents a novel polynomial-time algorithm that guarantees each agent at least 2/3 of their maximin share, advancing prior constant-agent methods.
• It achieves a 7/8-approximation for three-agent scenarios and an exact solution for valuations in {0, 1, 2}, showcasing targeted improvements.
• The research underscores the potential of algorithmic fairness in AI, using bipartite matching and probabilistic analysis to enhance equitable resource allocation.

Analyzing Approximation Algorithms for Maximin Share Allocations

The paper under consideration explores the computational challenges and solutions associated with maximin share allocations, a novel notion of fairness within the broader domain of fair division problems. This concept is particularly geared towards scenarios involving the allocation of indivisible goods among multiple agents, and it arises from the inability to guarantee fair allocations under traditional notions such as envy-freeness and proportionality. Maximin share allocations aim to provide each agent with the greatest value they can secure for themselves when partitioning the goods, thereby receiving their least preferred bundle under worst-case assumptions.

Core Contributions and Results

The primary contribution of the paper is a polynomial time algorithm that achieves a $\frac{2}{3} - \varepsilon$ approximation for the maximin share allocations, ensuring that each agent receives a bundle worth at least $\frac{2}{3}$ of their maximin share value. This improves upon previous work by \citet{PW14} that developed an algorithm with the same approximation ratio, but only efficient for instances with a constant number of agents. The novel solution leverages strategic adjustments in algorithm design, including the construction of matchings in a bipartite graph representation of the allocation problem.

In addition to the main result, the paper presents further findings for specific cases:

1. Three Agents: The paper introduces an algorithm that guarantees a $\frac{7}{8}$-approximation for instances with exactly three agents. This represents a significant improvement over previous $\frac{3}{4}$-approximation results for the same scenario.
2. Restricted Valuation Domains: When item values are restricted to $\{0, 1, 2\}$, the authors exhibit an exact algorithm, effectively eliminating any approximation loss, and extending previous findings for binary valuations.

Theoretical Implications and Challenges

The research addresses the inherent difficulty of achieving fair allocations in discrete settings, providing substantial insights into the feasibility of maximin share allocations and their approximations. The authors' probabilistic analysis further reinforces the concept's viability, demonstrating that maximin share allocations are likely to exist in randomly generated instances with uniform valuations.

One challenging aspect of the domain remains the perfecting of approximation ratios and understanding the boundaries for impossibility results. Whilst impressive advances have been made, there is no known absolute impossibility bound for such allocations, suggesting further intricate constructions are needed to illuminate the extremities of inapproximability.

Future Directions in AI

Speculating on the future development, the approach spearheaded by the authors could spur advancements in algorithmic fairness within AI, particularly in autonomous systems tasked with resource allocation under constraints. As AI systems become more involved in decision-making processes akin to human social dilemmas, concepts like maximin share offer a guideline balancing efficiency and equity. The integration of probabilistic analyses assures robustness in unpredictable or dynamic environments, enabling systems to align more closely with human-centric fairness standards.

In summary, the paper provides a comprehensive and mathematical exploration of a fairness concept that is particularly pertinent to modern computational needs. The numerical benchmarks demonstrated within offer a solid foundation on which future research can build, potentially paving the way to both theoretical expansions and practical implementations that enhance equitable decision-making processes in AI-driven contexts.