Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ -1)-approximation of envy-freeness up to any good (efx) and a 2 φ+2 -approximation of groupwise maximin share fairness (gmms), where φ is the golden ratio (φ ≈ 1.618). The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (ef1) and a 2/3-approximation of pairwise maximin share fairness (pmms). While efx is our primary focus, we also exhibit how to fine-tune our algorithm and further improve the guarantees for gmms or pmms. Finally, we show that gmms-and thus pmms and efx-allocations always exist when the number of goods does not exceed the number of agents by more than two. Best known (poly-time) 0.5 1 0.5 0.781 Algorithm 3 0.618 1 0.553 0.667 Variant in Thm. 4.7 0.6 1 0.571 0.667 Variant in Thm. 4.10 0.618 0.894 0.553 0.717 Table 1: Summary of our results and state of the art. Known results in the first row are due to Plaut and Roughgarden [2018], Lipton et al. [2004], Barman et al. [2018a], and Kurokawa [2017], respectively. At a technical level, our results are making use of two algorithms that are known to produce only ef1 allocations. The first one is a simple draft algorithm and the second one is the envycycle-elimination algorithm of Lipton et al. [2004] . Although these algorithms on their own