Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy

Xiaowei Wu; Shengwei Zhou

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arXiv · 2404.07707

Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy

Xiaowei Wu·Shengwei Zhou

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Abstract

We consider the problem of allocating $m$ indivisible items to a set of $n$ heterogeneous agents, aiming at computing a proportional allocation by introducing subsidy (money). It has been shown by Wu et al. (WINE 2023) that when agents are unweighted a total subsidy of $n/4$ suffices (assuming that each item has value/cost at most $1$ to every agent) to ensure proportionality. When agents have general weights, they proposed an algorithm that guarantees a weighted proportional allocation requiring a total subsidy of $(n-1)/2$, by rounding the fractional bid-and-take algorithm. In this work, we revisit the problem and the fractional bid-and-take algorithm. We show that by formulating the fractional allocation returned by the algorithm as a directed tree connecting the agents and splitting the tree into canonical components, there is a rounding scheme that requires a total subsidy of at most $n/3 - 1/6$.

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BibTeX RIS

Xiaowei Wu, Shengwei Zhou. 2024-04-11. Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy. https://arxiv.org/abs/2404.07707

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