Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy
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 $(n1)/2$, by rounding the fractional bidandtake algorithm. In this work, we revisit the problem and the fractional bidandtake 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 $n/3  1/6$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.07707
 arXiv:
 arXiv:2404.07707
 Bibcode:
 2024arXiv240407707W
 Keywords:

 Computer Science  Computer Science and Game Theory
 EPrint:
 30 pages, 11 figures