The classic cake-cutting problem provides a model for addressing fair and efficient allocation of a divisible, heterogeneous resource (metaphorically, the cake) among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake, this work establishes algorithmic and hardness results for multiple fairness/efficiency measures.First, we consider the well-studied notion of envy-freeness and develop an efficient algorithm that finds a cake division (with connected pieces) wherein the envy is multiplicatively within a factor of 3+o(1). The same algorithm in fact achieves an approximation ratio of 3+o(1) for the problem of finding cake divisions with as large a Nash social welfare (NSW) as possible. NSW is another standard measure of fairness and this work also establishes a connection between envy-freeness and NSW: approximately envy-free cake divisions (with connected pieces) always have near-optimal Nash social welfare. Furthermore, we develop an approximation algorithm for maximizing the ρ-mean welfarethis unifying objective, with different values of ρ, interpolates between notions of fairness (NSW) and efficiency (average social welfare). Finally, we complement these algorithmic results by proving that maximizing NSW (and, in general, the ρ-mean welfare) is APX-hard in the cake-division context.

Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a multiplier between 0 and 1, and her bid on each item is multiplicatively scaled down by the multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is the pacing equilibrium.In this work, we show that the problem of nding an approximate pacing equilibrium is PPAD-complete for second-price auctions. This resolves an open question of [2]. As a consequence of our hardness result, we show that the tatonnement-style budget-management dynamics introduced by [1] is unlikely to converge e ciently for repeated second-price auctions. Our hardness result also implies the existence of a re nement of supply-aware market equilibria which is hard to compute with simple linear utilities.

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Single-leg revenue management is a foundational problem of revenue management that has been particularly impactful in the airline and hotel industry: Given n units of a resource, e.g. flight seats, and a stream of sequentially-arriving customers segmented by fares, what is the optimal online policy for allocating the resource. Previous work focused on designing algorithms when forecasts are available, which are not robust to inaccuracies in the forecast, or online algorithms with worst-case performance guarantees, which can be too conservative in practice. In this work, we look at the single-leg revenue management problem through the lens of the algorithms-withadvice framework, which attempts to optimally incorporate advice/predictions about the future into online algorithms. In particular, we characterize the Pareto frontier that captures the tradeoff between consistency (performance when advice is accurate) and competitiveness (performance when advice is inaccurate) for every advice. Moreover, we provide an online algorithm that always achieves performance on this Pareto frontier. We also study the class of protection level policies, which is the most widely-deployed technique for single-leg revenue management: we provide an algorithm to incorporate advice into protection levels that optimally trades off consistency and competitiveness. Moreover, we empirically evaluate the performance of these algorithms on synthetic data. We find that our algorithm for protection level policies performs remarkably well on most instances, even if it is not guaranteed to be on the Pareto frontier in theory.