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In this paper, we consider the classic fair division problem of allocating m divisible items to n agents with linear valuations over the items. We define novel notions of fair shares from the perspective of individual agents via the cake-cutting process. These shares generalize the notion of proportionality by taking into account the valuations of other agents via constraints capturing envy. We study what fraction (approximation) of these shares are achievable in the worst case, and present tight and non-trivial approximation bounds as a function of n and m. In particular, we show a tight approximation bound of Θ(√n) for various notions of such shares. We show this bound via a novel application of dual fitting, which may be of independent interest. We also present a bound of O(m^(2/3)) for a strict notion of share, with an almost matching lower bound. We further develop weaker notions of shares whose approximation bounds interpolate smoothly between proportionality and the shares described above. We finally present empirical results showing that our definitions lead to more reasonable shares than the standard fair share notion of proportionality. 

We consider the setting where a user with sensitive features wishes to obtain a recommendation from a server in a differentially private fashion. We propose a "multi-selection" architecture where the server can send back multiple recommendations and the user chooses one from these that matches best with their private features. When the user feature is one-dimensional - on an infinite line - and the accuracy measure is defined w.r.t some increasing function 𝔥(.) of the distance on the line, we precisely characterize the optimal mechanism that satisfies differential privacy. The specification of the optimal mechanism includes both the distribution of the noise that the user adds to its private value, and the algorithm used by the server to determine the set of results to send back as a response. We show that Laplace is an optimal noise distribution in this setting. Furthermore, we show that this optimal mechanism results in an error that is inversely proportional to the number of results returned when the function 𝔥(.) is the identity function. 

We consider the problem of assigning students to schools when students have different utilities for schools and schools have limited capacities. The students belong to demographic groups, and fairness over these groups is captured either by concave objectives, or additional constraints on the utility of the groups. We present approximation algorithms for this assignment problem with group fairness via convex program rounding. These algorithms achieve various trade-offs between capacity violation and running time. We also show that our techniques easily extend to the setting where there are arbitrary constraints on the feasible assignment, capturing multi-criteria optimization. We present simulation results that demonstrate that the rounding methods are practical even on large problem instances, with the empirical capacity violation being much better than the theoretical bounds.