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We study fair distribution of a collection of m indivisible goods among a group of n agents, using the widely recognized fairness principles of Maximin Share (MMS) and Any Price Share (APS). These principles have undergone thorough investigation within the context of additive valuations. We explore these notions for valuations that extend beyond additivity.First, we study approximate MMS under the separable (piecewise-linear) concave (SPLC) valuations, an important class generalizing additive, where the best known factor was 1/3-MMS. We show that 1/2-MMS allocation exists and can be computed in polynomial time, significantly improving the state-of-the-art.We note that SPLC valuations introduce an elevated level of intricacy in contrast to additive. For instance, the MMS value of an agent can be as high as her value for the entire set of items. We use a relax-and-round paradigm that goes through competitive equilibrium and LP relaxation. Our result extends to give (symmetric) 1/2-APS, a stronger guarantee than MMS.APS is a stronger notion that generalizes MMS by allowing agents with arbitrary entitlements. We study the approximation of APS under submodular valuation functions. We design and analyze a simple greedy algorithm using concave extensions of submodular functions. We prove that the algorithm gives a 1/3-APS allocation which matches the best-known factor. Concave extensions are hard to compute in polynomial time and are, therefore, generally not used in approximation algorithms. Our approach shows a way to utilize it within analysis (while bypassing its computation), and hence might be of independent interest. 

We study the problem of approximating maximum Nash social welfare (NSW) when allocatingmindivisible items amongnasymmetric agents with submodular valuations. TheNSWis a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents’ valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with a factor independent ofmwas known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations before this work. In this article, we extend our understanding of theNSWproblem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high-valued items are done separately by un-matching certain items and re-matching them by different processes in both algorithms. We show that these approaches achieve approximation factors ofO(n) andO(nlogn) for additive and submodular cases, independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1). Furthermore, we show that theNSWproblem under submodular valuations is strictly harder than all currently known settings with an\(\frac{\mathrm{e}}{\mathrm{e}-1}\)factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of\(\frac{\mathrm{e}}{\mathrm{e}-1}\), hence resolving it completely.