Tree Splitting Based Rounding Scheme for Weighted Proportional Allocations with Subsidy

TL;DR

This work tackles fair allocation of $m$ indivisible items among $n$ weighted agents with subsidies to ensure weighted proportionality (WPROPS). It introduces a two-stage method: compute a fractional WPROP allocation via the Fractional Bid-and-Take Algorithm (FBTA) and then apply a tree-splitting rounding framework on the induced item-sharing graph to obtain an integral allocation with bounded subsidies. The core technical advance is showing that the FBTA output forms a directed tree, enabling a principled rounding by splitting the tree into canonical components; this yields a total subsidy bound of at most $\frac{n}{3}-\frac{1}{6}$ for chores (and $\frac{n}{3}$ for goods). A key challenge is handling shattered items (shared by three or more agents), for which the authors develop atom-path and expanded-atom-path constructs to maintain consistency during rounding. Overall, the paper closes a gap between the unweighted $n/4$ bound and the earlier weighted $(n-1)/2$ bound, and introduces a versatile graph-based framework that may extend to other fair-division settings with monetary transfers.

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 $n/3 - 1/6$.

Figures (11)

• Figure 1: When simply viewing items as edges between agents, the graph might be complex when an item is shared by more than two agents, e.g., $e_2$ is (fractional) allocated to agents $2,3,4$.
• Figure 2: The item-sharing graph for the returned allocation on instance $\mathcal{I}^*$.
• Figure 3: The item-sharing graph $G^*$ of instance $\mathcal{I}^*$ is a tree with root of agent $6$.
• Figure 4: A simple splitting of the item-sharing graph of instance $\mathcal{I}^*$.
• Figure 5: The item-sharing graph for three agents.

Theorems & Definitions (66)

• Definition 2.1: WPROP
• Definition 2.2: WPROPS
• Definition 2.3: Identical Ordering (IDO) Instances
• Lemma 2.4
• Lemma 3.1: conf/wine/WuZZ23
• Example 3.2
• Definition 3.3: successor
• Theorem 3.5
• proof
• Theorem 4.1