Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy
<|reference_start|>Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy: 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 $n/3 - 1/6$.<|reference_end|>
@article{wu2024tree,
    title={Tree Splitting Based Rounding Scheme for Weighted Proportional
  Allocations with Subsidy},
    author={Xiaowei Wu, Shengwei Zhou},
    journal={arXiv preprint arXiv:2404.07707},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.07707},
    primaryClass={cs.GT}
}