Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy

TL;DR

This paper introduces a new tree-based rounding scheme for weighted proportional allocations with subsidies, reducing the total subsidy needed compared to previous algorithms.

Contribution

It presents a novel tree splitting rounding scheme that improves subsidy bounds for weighted proportional allocations.

Findings

The new scheme requires at most n/3 - 1/6 total subsidy.

It generalizes and improves upon previous rounding algorithms.

The approach uses a directed tree structure for fractional allocations.

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 $n/3 - 1/6$.