1/2 Approximate MMS Allocation for Separable Piecewise Linear Concave Valuations

TL;DR

This paper improves the approximation factor for fair division under complex valuations, showing a 1/2-MMS allocation for SPLC valuations and a 1/3-APS for submodular valuations, using novel relax-and-round and greedy algorithms.

Contribution

It introduces polynomial-time algorithms achieving better approximation guarantees for MMS and APS under non-additive valuations, extending fair division theory.

Findings

1/2-MMS allocation exists and is computable in polynomial time for SPLC valuations.

A simple greedy algorithm achieves 1/3-APS for submodular valuations.

The relax-and-round paradigm extends to complex valuation classes, improving fairness guarantees.

Abstract

We study fair distribution of a collection of m indivisible goods among a group of n agents, using the widely recognized fairness principles of Maximin Share (MMS) and Any Price Share (APS). These principles have undergone thorough investigation within the context of additive valuations. We explore these notions for valuations that extend beyond additivity. First, we study approximate MMS under the separable (piecewise-linear) concave (SPLC) valuations, an important class generalizing additive, where the best known factor was 1/3-MMS. We show that 1/2-MMS allocation exists and can be computed in polynomial time, significantly improving the state-of-the-art. We note that SPLC valuations introduce an elevated level of intricacy in contrast to additive. For instance, the MMS value of an agent can be as high as her value for the entire set of items. Further, the equilibrium computation problem, which is polynomial-time for additive valuations, becomes intractable for SPLC. We use a relax-and-round paradigm that goes through competitive equilibrium and LP relaxation. Our result extends to give (symmetric) 1/2-APS, a stronger guarantee than MMS. APS is a stronger notion that generalizes MMS by allowing agents with arbitrary entitlements. We study the approximation of APS under submodular valuation functions. We design and analyze a simple greedy algorithm using concave extensions of submodular functions. We prove that the algorithm gives a 1/3-APS allocation which matches the current best-known factor. Concave extensions are hard to compute in polynomial time and are, therefore, generally not used in approximation algorithms. Our approach shows a way to utilize it within analysis (while bypassing its computation), and might be of independent interest.