We study the sublinear multivariate mean estimation problem in $d$-dimensional Euclidean space. Specifically, we aim to find the mean $\mu$ of a ground point set $A$, which minimizes the sum of squared Euclidean distances of the points in $A$ to $\mu$. We first show that a multiplicative $(1+\varepsilon)$ approximation to $\mu$ can be found with probability $1-\delta$ using $O(\varepsilon^{-1}\log \delta^{-1})$ many independent uniform random samples, and provide a matching lower bound. Furthermore, we give two estimators with optimal sample complexity that can be computed in optimal running time for extracting a suitable approximate mean: 1. The coordinate-wise median of $\log \delta^{-1}$ sample means of sample size $\varepsilon^{-1}$. As a corollary, we also show improved convergence rates for this estimator for estimating means of multivariate distributions. 2. The geometric median of $\log \delta^{-1}$ sample means of sample size $\varepsilon^{-1}$. To compute a solution efficiently, we design a novel and simple gradient descent algorithm that is significantly faster for our specific setting than all other known algorithms for computing geometric medians. In addition, we propose an order statistics approach that is empirically competitive with these algorithms, has an optimal sample complexity and matches the running time up to lower order terms. We finally provide an extensive experimental evaluation among several estimators which concludes that the geometric-median-of-means-based approach is typically the most competitive in practice.

Motivated by recent work in computational social choice, we extend the metric distortion framework to clustering problems. Given a set of n agents located in an underlying metric space, our goal is to partition them into k clusters, optimizing some social cost objective. The metric space is defined by a distance function d between the agent locations. Information about d is available only implicitly via n rankings, through which each agent ranks all other agents in terms of their distance from her. Still, even though no cardinal information (i.e., the exact distance values) is available, we would like to evaluate clustering algorithms in terms of social cost objectives that are defined using d. This is done using the notion of distortion, which measures how far from optimality a clustering can be, taking into account all underlying metrics that are consistent with the ordinal information available. Unfortunately, the most important clustering objectives (e.g., those used in the well-known k-median and k-center problems) do not admit algorithms with finite distortion. To sidestep this disappointing fact, we follow two alternative approaches: We first explore whether resource augmentation can be beneficial. We consider algorithms that use more than k clusters but compare their social cost to that of the optimal k-clusterings. We show that using exponentially (in terms of k) many clusters, we can get low (constant or logarithmic) distortion for the k-center and k-median objectives. Interestingly, such an exponential blowup is shown to be necessary. More importantly, we explore whether limited cardinal information can be used to obtain better results. Somewhat surprisingly, for k-median and k-center, we show that a number of queries that is polynomial in k and only logarithmic in n (i.e., only sublinear in the number of agents for the most relevant scenarios in practice) is enough to get constant distortion.

We study the fair allocation of indivisible goods among agents with identical, additive valuations but individual budget constraints. Here, the indivisible goods--each with a specific size and value--need to be allocated such that the bundle assigned to each agent is of total size at most the agent's budget. Since envy-free allocations do not necessarily exist in the indivisible goods context, compelling relaxations--in particular, the notion of envy-freeness up to k goods (EFk)--have received significant attention in recent years. In an EFk allocation, each agent prefers its own bundle over that of any other agent, up to the removal of k goods, and the agents have similarly bounded envy against the charity (which corresponds to the set of all unallocated goods). It has been shown in prior work that an allocation that satisfies the budget constraints and maximizes the Nash social welfare is 1/4-approximately EF1. However, the computation (or even existence) of exact EFk allocations remained an intriguing open problem. We make notable progress towards this by proposing a simple, greedy, polynomial-time algorithm that computes EF2 allocations under budget constraints. Our algorithmic result implies the universal existence of EF2 allocations in this fair division context. The analysis of the algorithm exploits intricate structural properties of envy-freeness. Interestingly, the same algorithm also provides EF1 guarantees for important special cases. Specifically, we settle the existence of EF1 allocations for instances in which: (i) the value of each good is proportional to its size, (ii) all the goods have the same size, or (iii) all the goods have the same value. Our EF2 result even extends to the setting wherein the goods' sizes are agent specific.

We study fair division of goods under the broad class of generalized assignment constraints. In this constraint framework, the sizes and values of the goods are agent-specific, and one needs to allocate the goods among the agents fairly while further ensuring that each agent receives a bundle of total size at most the corresponding budget of the agent. Since, in such a constraint setting, it may not always be feasible to partition all the goods among the agents, we conform---as in recent works---to the construct of charity to designate the set of unassigned goods. For this allocation framework, we obtain existential and computational guarantees for envy-free (appropriately defined) allocation of divisible and indivisible goods, respectively, among agents with individual, additive valuations for the goods.