Quantifying the Welfare Loss from Connectivity Constraints in Resource Allocation

Core Concepts

The worst-case degradation of social welfare when allocating connected subsets of items is quantified through the concepts of egalitarian and utilitarian price of connectivity.

Abstract

The paper studies the allocation of indivisible goods that form an undirected graph and investigates the worst-case welfare loss when requiring that each agent must receive a connected subgraph. The focus is on both egalitarian and utilitarian welfare.
Key highlights:

The authors introduce the concepts of egalitarian and utilitarian price of connectivity, which capture the worst-case ratio between the optimal welfare and the optimal connected welfare.
For the two-agent case, they provide tight or asymptotically tight bounds on the price of connectivity for various graph classes, including complete graphs with a matching removed, complete bipartite graphs, and graphs with connectivity 1 or 2.
For the three-agent case, they show that the egalitarian price of connectivity for tree graphs is equal to the maximum number of disjoint connected subgraphs when two vertices are removed.
They also extend their results to the general case with any number of agents for certain graph classes like stars, paths, and cycles.
The results are supplemented with algorithms that find connected allocations with a welfare guarantee corresponding to the price of connectivity.

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Key Insights Distilled From

Welfare Loss in Connected Resource Allocation

by Xiaohui Bei,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03467.pdf

Deeper Inquiries

How do the egalitarian and utilitarian prices of connectivity compare across different graph classes

The egalitarian and utilitarian prices of connectivity vary across different graph classes based on their structural properties. In general, the egalitarian price of connectivity tends to be higher than the utilitarian price of connectivity for most graph classes. This is because the egalitarian welfare focuses on the worst-off agent, leading to a more conservative measure of fairness compared to the utilitarian welfare, which considers the sum of all agents' utilities.
For certain graph classes, such as complete graphs with a non-empty matching removed, the egalitarian price of connectivity can be strictly higher than 1, indicating a significant welfare loss when imposing connectivity constraints. On the other hand, the utilitarian price of connectivity may remain closer to 1 for the same graph classes, reflecting a more balanced distribution of welfare among agents.
While there may not be a direct general relationship between the egalitarian and utilitarian prices of connectivity across all graph classes, the comparison highlights the trade-off between fairness and efficiency in resource allocation scenarios with connectivity constraints.

Can we establish any general relationships between them

The computational complexity of finding the optimal connected allocation that maximizes egalitarian or utilitarian welfare can vary depending on the graph structure and the number of agents involved. In the context of the presented algorithms in the research work, the time complexity is influenced by the size of the graph, the number of agents, and the specific constraints imposed.
The algorithms provided in the study offer polynomial-time solutions for finding connected allocations with guaranteed welfare corresponding to the price of connectivity. However, the complexity may increase for larger graphs or a higher number of agents, especially when considering more intricate graph classes or connectivity constraints.
To achieve better time complexity, the algorithms can be optimized by incorporating heuristic approaches, parallel processing techniques, or leveraging data structures that expedite the search for optimal allocations. Additionally, exploring approximation algorithms or refining the algorithm design based on specific graph characteristics can help enhance efficiency in finding connected allocations that optimize welfare metrics.

What is the computational complexity of finding the optimal connected allocation that maximizes egalitarian or utilitarian welfare

The concepts of price of connectivity can be generalized to various resource allocation problems beyond the graph-theoretic setting, particularly in scenarios with connectivity constraints such as multi-robot task allocation or network design. In these contexts, the price of connectivity can quantify the trade-off between collaborative benefits from connected allocations and the welfare loss incurred due to connectivity constraints.
For multi-robot task allocation, enforcing connectivity constraints can enhance coordination and communication among robots, leading to improved task performance and efficiency. However, this may come at the cost of reduced overall welfare if the connectivity constraints limit the allocation possibilities. By calculating the price of connectivity in this domain, decision-makers can evaluate the impact of connectivity requirements on welfare outcomes and make informed decisions.
In network design, connectivity constraints play a crucial role in ensuring robustness and reliability of communication networks. Analyzing the price of connectivity in network design problems can help optimize resource allocation strategies to balance connectivity requirements with welfare considerations, ultimately leading to more resilient and efficient network architectures.