[Paper Review] Ordinal Maximin Share Approximation for Chores

Seyed Hossein Hosseini, Andrew Searns+1 more|arXiv (Cornell University)|Jan 19, 2022

Auction Theory and Applications4 citations

TL;DR

This paper presents a polynomial-time algorithm for achieving a 1-out-of-⌊2n/3⌋ maximin share (MMS) approximation in fair division of indivisible chores, and proves the existence of a 1-out-of-⌊3n/4⌋ MMS allocation. It further develops a practical algorithm that approximates the ⌊3n/4⌋ bound up to a logarithmic factor using bin-packing techniques.

ABSTRACT

We study the problem of fairly allocating a set of m indivisible chores (items with non-positive value) to n agents. We consider the desirable fairness notion of 1-out-of-d maximin share (MMS) -- the minimum value that an agent can guarantee by partitioning items into d bundles and receiving the least valued bundle -- and focus on ordinal approximation of MMS that aims at finding the largest d <= n for which 1-out-of-d MMS allocation exists. Our main contribution is a polynomial-time algorithm for 1-out-of-floor(2n/3) MMS allocation, and a proof of existence of 1-out-of-floor(3n/4) MMS allocation of chores. Furthermore, we show how to use recently-developed algorithms for bin-packing to approximate the latter bound up to a logarithmic factor in polynomial time.

Motivation & Objective

• To address the challenge of fair allocation of indivisible chores among agents using ordinal MMS approximation, a fairness criterion that generalizes the cut-and-choose protocol to indivisible items.
• To overcome the limitations of multiplicative MMS approximations by introducing an ordinal approach that provides robust, interpretable fairness guarantees based on bundle rankings.
• To develop efficient algorithms that achieve strong ordinal MMS approximations for chores, particularly focusing on improving the best-known approximation ratio.
• To bridge theoretical existence results with practical computation by leveraging bin-packing algorithms to approximate the optimal ordinal MMS bound within a logarithmic factor.

Proposed method

• Proposes a greedy algorithm that achieves a 1-out-of-⌊2n/3⌋ MMS approximation by carefully bounding the number of large chores and using a novel partitioning strategy.
• Employs a novel technical lemma (Lemma 3.8) to upper-bound the number of large chores, which is critical for handling the constraints of chore allocation where all items must be assigned.
• Proves the existence of a 1-out-of-⌊3n/4⌋ MMS allocation using exact computation of MMS values, despite the intractability of computing such values in general.
• Adapts the harmonic algorithm (HR) for bin-packing to approximate the MMS threshold: each chore is treated as an item of size |v_i(c)|, and bin size corresponds to the MMS threshold.
• Uses binary search over possible bin sizes to compute a threshold β_i for each agent i, ensuring that the agent’s allocation value is at least β_i, which is shown to be a lower bound on the MMS for d = ⌊3n/4⌋ - a·log(⌊3n/4⌋).
• Combines Algorithm 1 (a threshold-based allocation) with computed thresholds from bin-packing to yield a complete, polynomial-time allocation that guarantees 1-out-of-⌊⌊3n/4⌋ - a·log(⌊3n/4⌋)⌋ MMS for all agents.

Experimental results

Research questions

• RQ1 Can a polynomial-time algorithm achieve a 1-out-of-⌊2n/3⌋ MMS approximation for chore allocation, despite the failure of standard goods allocation heuristics?
• RQ2 Does a 1-out-of-⌊3n/4⌋ MMS allocation exist for chores, even when exact MMS computation is infeasible for large instances?
• RQ3 Can the theoretical existence of a 1-out-of-⌊3n/4⌋ MMS allocation be transformed into a practical, polynomial-time algorithm with bounded approximation loss?
• RQ4 How does the ordinal MMS approximation framework compare to multiplicative MMS approximations in terms of robustness and computational feasibility for chore allocation?

Key findings

• A polynomial-time algorithm is presented that guarantees a 1-out-of-⌊2n/3⌋ MMS allocation for chores, with runtime strongly polynomial in the number of agents and chores.
• The existence of a 1-out-of-⌊3n/4⌋ MMS allocation for chores is proven, establishing the best known theoretical ordinal approximation ratio for this problem.
• An efficient algorithm is developed that approximates the ⌊3n/4⌋ MMS bound up to a logarithmic factor, achieving 1-out-of-⌊⌊3n/4⌋ - a·log(⌊3n/4⌋)⌋ MMS for some fixed a > 0.
• The algorithm runs in time polynomial in the binary representation of the input values, making it practical for real-world applications with integer-valued chore utilities.
• The approach is robust: the ordinal approximation is less sensitive to small changes in valuation profiles than multiplicative approximations, offering stronger conceptual fairness justification.
• The method leverages bin-packing algorithms and binary search to compute thresholds that are provably lower bounds on the MMS value for a given d, ensuring completeness and fairness of the allocation.