Pushing the Frontier on Approximate EFX Allocations
Georgios Amanatidis, Aris Filos-Ratsikas, Alkmini Sgouritsa
TL;DR
The paper tackles the problem of allocating indivisible goods to agents with additive valuations with the goal of approximate envy-freeness up to any good, achieving a $2/3$-EFX guarantee in three key settings. It introduces the Property-Preserving Partial Allocation (3PA) framework and a suite of envy-graph based techniques to iteratively refine partial allocations until a complete $2/3$-EFX allocation is obtained in polynomial time. The main contributions are (i) a polynomial-time $2/3$-EFX existence result for multigraph-value instances and for up to seven agents, (ii) a $2/3$-EFX result for 3-value instances with multiple algorithmic variants that handle different parameter regimes, and (iii) a refined approach (3PA$^{++}$) that addresses the hardest subcase $b>1/2$ with $b+c<2/3$. These results push the frontier beyond the known $ heta=0.618$ barrier and provide a versatile framework that could guide future work toward exact EFX or broader classes of valuations. The techniques—combining partial allocations, multiple envy graphs, and cycle/path resolutions—offer a scalable pathway for designing fair allocations in restricted, but practically relevant, settings. The work also highlights the structural challenges of exact EFX and offers insights that may inform extensions to chores or more general additive valuations.
Abstract
We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($α$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $α$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.