The Computational Complexity of Envy-Free Allocations with Externalities

Core Concepts

Determining the computational complexity of finding envy-free allocations in the presence of externalities, where agents' valuations depend on the entire allocation, not just their own bundle.

Abstract

The content explores the computational complexity of fairly allocating a set of indivisible items under externalities. In this setting, in addition to the utility an agent gets from their own bundle, they also receive utility from items allocated to other agents. The authors focus on the extended definitions of envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX), and provide a comprehensive analysis of their complexity in several different scenarios. Key highlights: The authors prove that it is NP-complete to decide whether there exists an EFX allocation, even when there are only three agents, or even when there are only six different values for the items. They complement these negative results by showing that when both the number of agents and the number of different values for items are bounded by a parameter, the problem becomes fixed-parameter tractable. The authors also prove that two-valued and binary-valued instances are equivalent and that EFX and EF1 allocations coincide for this class of instances. Motivated by real-life scenarios, the authors focus on a class of structured valuation functions, termed agent/item-correlated, and prove their equivalence to the "standard" setting without externalities.

Stats

There are 2n integers in the sequence S = (s1, s2, ..., s2n) for some n ∈ N. M = (smax - smin) * n^2, where smin and smax are the minimum and maximum integers in S, respectively. B = 1/2 * Σi∈[2n] (si - smin).

Quotes

"Typically, an instance of the fair division problem consists of a set of indivisible items, and a set of agents each of whom has their own valuation function. The task is to partition the items into bundles and allocate each bundle to an agent such that from the point of view of every agent this allocation is 'fair'." "Motivated by real-life scenarios like the two above, Aziz et al. [2023b] recently proposed a new model suitable to capture the situations where externalities occur; interestingly, for divisible items, the first models that incorporate externalities were proposed many years ago [Brânzei et al., 2013; Li et al., 2015]."

Key Insights Distilled From

The Complexity of Fair Division of Indivisible Items with Externalities

by Argy... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2308.08869.pdf

Deeper Inquiries

How can the results on agent/item-correlated valuations be extended to more general classes of structured valuations

The results on agent/item-correlated valuations can be extended to more general classes of structured valuations by considering different types of correlations between agents and items. In the context of fair division with externalities, agent/item-correlated valuations capture scenarios where an agent's utility from an item depends not only on the item itself but also on the agent to whom the item is allocated. This correlation can be extended to include more complex relationships, such as group preferences, hierarchical structures, or conditional dependencies. By exploring these more general classes of structured valuations, researchers can analyze how different types of correlations impact the fairness and efficiency of allocations in the presence of externalities. Understanding the implications of various correlation structures can provide insights into designing allocation mechanisms that better reflect real-world scenarios where externalities play a significant role in decision-making processes.

What are the implications of the hardness results for the design of practical algorithms for fair division with externalities

The hardness results for fair division with externalities have significant implications for the design of practical algorithms in several ways. Firstly, these results highlight the computational complexity of finding fair allocations in settings with externalities, indicating that the problem is NP-complete even for relatively small instances. This complexity suggests that designing efficient algorithms for fair division with externalities requires careful consideration of the underlying computational challenges. Additionally, the hardness results can guide the development of approximation algorithms or heuristic approaches to tackle fair division problems with externalities. While exact solutions may be computationally infeasible for larger instances, approximation algorithms can provide near-optimal solutions within a reasonable time frame. By leveraging insights from the hardness results, researchers can focus on developing algorithmic techniques that balance computational efficiency with solution quality. Moreover, the hardness results underscore the importance of parameterized complexity in addressing computational challenges in fair division with externalities. By identifying tractable fragments of the problem based on specific parameters, researchers can tailor algorithmic approaches to handle instances that exhibit certain structural properties. This approach allows for a more nuanced understanding of the computational landscape of fair division problems and can lead to the development of specialized algorithms for different scenarios.

Can the parameterized complexity results be further improved or generalized to capture a wider range of practical scenarios

The parameterized complexity results in fair division with externalities can be further improved or generalized to capture a wider range of practical scenarios by considering additional parameters or refining existing parameterizations. One approach to enhancing the parameterized complexity analysis is to explore the impact of different combinations of parameters on the computational complexity of the problem. By investigating how multiple parameters interact and influence the complexity of fair division with externalities, researchers can identify more precise boundaries between tractable and intractable instances. Furthermore, generalizing the parameterized complexity results may involve extending the analysis to incorporate additional constraints or variations in the problem formulation. For example, considering different types of valuations, externalities, or allocation constraints can lead to a more comprehensive understanding of the computational complexity of fair division with externalities. By exploring a broader spectrum of scenarios and parameter settings, researchers can develop a more nuanced framework for analyzing the complexity of fair division problems and designing tailored algorithmic solutions.