Bad News for Couples: Tight Lower Bounds for Fair Division of Indivisible Items

We consider the problem of fairly allocating indivisible goods to couples, where each couple consists of two agents with distinct additive valuations. We show that there exist instances of allocating indivisible items to $n$ couples for which envy-freeness up to $\mathrm{\Omega }(\sqrt{n})$ items cannot be guaranteed. This closes the gap by matching the upper bound of Manurangsi and Suksompong, which applies to arbitrary instances with $n$ agents in total. This result is somewhat surprising, as that upper bound was conjectured not to be tight for instances consisting only of small groups.