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On Approximate MMS Allocations on Restricted Graph Classes

Created by
  • Haebom

Author

V aclav Bla\v{z}ej, Micha{\l} D\k{e}bski, Zbigniew Lonc, Marta Piecyk, Pawe{\l} Rz\k{a} zewski

Outline

This paper studies the problem of fair distribution of indivisible goods under connectivity constraints. Goods are represented as vertices in a connected graph, and the set of goods allocated to each agent is a connected subgraph of this graph. The paper focuses on the maximin share criterion, a widely studied fairness criterion. It is well known that there may not be an allocation that satisfies the min-max share criterion even without connectivity constraints. Therefore, it is natural to find an approximate allocation that guarantees each agent a connected bundle of goods that is at least a certain percentage of the min-max share value. It is known that such approximate allocations exist for some graph classes, such as complete graphs, cyclic graphs, and d-claw-free graphs for fixed d. However, whether such approximate allocations exist for all graph classes is an open question. In this paper, we continue our systematic study of the existence of approximate allocations for limited graph classes and show that such allocations exist for several well-studied graph classes, including block graphs, cactus graphs, complete multipartite graphs, and partitioned graphs.

Takeaways, Limitations

Takeaways: Deepened our understanding of the fair distribution problem of connected goods by proving the existence of approximate allocations that satisfy the min-max sharing criterion under connectivity constraints in various graph types, such as block graphs, cactus graphs, complete multipartite graphs, and partitioned graphs.
Limitations: The existence of approximate assignments for all graph types remains an open question. Research on graph types other than those proven is needed. Algorithm design and efficiency analysis for practical applications are lacking.
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