[Paper Review] The Fair Division of Hereditary Set Systems

Zhentao Li, Adrian Vetta|arXiv (Cornell University)|Dec 22, 2018

Game Theory and Voting Systems16 references4 citations

TL;DR

This paper presents a polynomial-time algorithm for fair division in hereditary set systems, where items are allocated such that each agent receives a bundle valued at least 0.3636 times their maximin share. The method uses iterative approximation with valuation queries and improves upon the prior 0.2 guarantee, with analysis showing the bound is nearly tight at 0.3738.

ABSTRACT

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least $0.3636$ times the maximin share of the agent. This improves upon the current best known guarantee of $0.2$ due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most $0.3738$. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.

Motivation & Objective

• To address fair allocation of indivisible items in hereditary set systems, where feasibility constraints limit which item subsets are allowed.
• To improve the best-known approximation guarantee for maximin share fairness beyond the prior 0.2 bound.
• To design a polynomial-time algorithm using only valuation oracles, enabling practical deployment.
• To analyze the tightness of the approximation guarantee through a nearly matching upper bound instance.

Proposed method

• The algorithm uses a binary search-like approach on the maximin share value, iteratively refining estimates using valuation queries.
• It maintains a set of candidate bundles and applies a removal procedure to prune items that cannot be part of a high-value feasible allocation.
• A key subroutine evaluates whether any item can be removed from a current bundle without violating feasibility or value thresholds.
• The method leverages the hereditary property to ensure that all subsets of feasible bundles remain feasible, simplifying the search space.
• It employs a multiplicative approximation factor (1−δ) to progressively narrow the estimate of each agent’s maximin share.
• The algorithm is implemented in polynomial time by bounding the number of iterations using logarithmic scaling on the value ratio between upper and lower bounds of the MMS.

Experimental results

Research questions

• RQ1 Can a better approximation guarantee than 0.2 be achieved for fair division in hereditary set systems under the maximin share criterion?
• RQ2 Is there a polynomial-time algorithm that achieves improved fairness guarantees using only valuation oracles?
• RQ3 How close is the performance of the proposed algorithm to the theoretical optimum, and can this be quantified?
• RQ4 Does the algorithm’s performance degrade significantly in structured subclasses like matroids, or can the augmentation property be exploited for better guarantees?
• RQ5 Can the algorithm be adapted to handle additional constraints such as union feasibility in matroid-based systems?

Key findings

• The proposed algorithm guarantees each agent a bundle with value at least 0.3636 times their individual maximin share, improving upon the prior 0.2 bound.
• An instance is constructed where the algorithm’s performance is at most 0.3738, showing the bound is nearly tight with a gap of only 0.0072.
• The algorithm runs in polynomial time using only valuation oracle queries, making it practically feasible for large-scale settings.
• The method achieves a (1−δ)·(4/11) approximation for any δ>0, with 4/11 ≈ 0.3636 being the limiting guarantee.
• The analysis confirms that the algorithm maintains a lower bound on the approximation ratio through iterative refinement and invariant preservation.
• The upper bound example is a partition matroid, indicating that the current guarantee may not be improvable without exploiting stronger structural properties like the augmentation axiom.