Approximating APS under Submodular and XOS Valuations with Binary Marginals

We study the problem of fairly dividing indivisible goods among a set of agents under the fairness notion of Any Price Share (APS). APS is known to dominate the widely studied Maximin share (MMS). Since an exact APS allocation may not exist, the focus has traditionally been on the computation of approximate APS allocations. [4] studied the problem under additive valuations, and asked (i) how large can the APS value be compared to the MMS value? and (ii) what guarantees can one achieve beyond additive functions. We partly answer these questions by considering valuations beyond additive, namely submodular and XOS functions, with binary marginals. For the submodular functions with binary marginals, also known as matroid rank functions (MRFs), we show that APS is exactly equal to MMS. Consequently, following [5] we show that an exact APS allocation exists and can be computed efficiently while maximizing the social welfare. Complementing this result, we show that it is NP-hard to compute the APS value within a factor of 5/6 for submodular valuations with three distinct marginals of {0, 1/2, 1}. We then consider binary XOS functions, which are immediate generalizations of binary submodular functions in the complement free hierarchy. In contrast to the MRFs setting, MMS and APS values are not equal under this case. Nevertheless, we can show that they are only a constant factor apart. In particular, we show that under binary XOS valuations, MMS ≤ APS ≤ 2 · MMS + 1. Further, we show that this is almost the tightest bound we can get using MMS, by giving an instance where APS ≥ 2 · MMS. The upper bound on APS, combined with [17], implies a 0.1222-approximation for APS under binary XOS valuations. And the lower bound implies the nonexistence of better than 0.5-APS even when agents have identical valuations, which is in sharp contrast to the guaranteed existence of exact MMS allocation when agent valuations are identical.