On the Hardness of Maximin Share Allocations

The maximin share (MMS) guarantee is a central fairness benchmark for allocating indivisible items. Since Kurokawa, Procaccia and Wang [EC’14, JACM’18] showed that exact MMS allocations need not exist, much work has studied existence and computation of approximate MMS allocations. In contrast, a basic complexity question posed more than a decade ago by Bouveret and Lemaître [JAAMAS’16] has remained unresolved: how hard is it to decide whether an exact MMS allocation exists? For additive valuations, Lonc and Truszczynski [JAIR’20] showed membership in ${\mathrm{\Delta }}_2^P$ (also known as $P^{NP}$), but no hardness result was known. For the more general class of 2-additive valuations, Bouveret and Lemaître established NP-hardness, leaving a substantial gap to the ${\mathrm{\Delta }}_2^P$ upper bound. Moreover, the (precise) complexity of MMS existence in additive and $k$-additive settings was posed as an open question. We make progress on all of these fronts: (1) For additive goods, we prove that deciding MMS existence is $D^P$-hard, giving the first hardness result for this longstanding problem. (2) For 2-additive valuations, we close the complexity gap by proving ${\mathrm{\Delta }}_2^P$-completeness on a class of instances of monotone submodular goods. To the best of our knowledge this is the first result of this kind. We also prove weak coNP-hardness for three agents, thereby establishing a precise dichotomy with the known existence guarantee for two agents; and strong coNP-hardness when the number of agents is unrestricted. Moreover, the strong hardness construction produces an inverse-polynomial gap in the optimal MMS approximation ratio, ruling out an FPTAS for approximating this ratio unless P=NP. Finally, we show that all these results for goods extend to the chores setting through a polynomial-time transformation that preserves MMS existence.