We study the problem of designing procurement auctions for the strategic uncapacitated facility location problem: a company needs to procure a set of facility locations in order to serve its customers and each facility location is owned by a strategic agent. Each owner has a private cost for providing access to their facility (e.g., renting it or selling it to the company) and needs to be compensated accordingly. The goal is to design truthful auctions that decide which facilities the company should procure and how much to pay the corresponding owners, aiming to minimize the total cost, i.e., the monetary cost paid to the owners and the connection cost suffered by the customers (their distance to the nearest facility). We evaluate the performance of these auctions using the frugality ratio. We first analyze the performance of the classic VCG auction in this context and prove that its frugality ratio is exactly 3. We then leverage the learning-augmented framework and design auctions that are augmented with predictions regarding the owners' private costs. Specifically, we propose a family of learning-augmented auctions that achieve significant payment reductions when the predictions are accurate, leading to much better frugality ratios. At the same time, we demonstrate that these auctions remain robust even if the predictions are arbitrarily inaccurate, and maintain reasonable frugality ratios even under adversarially chosen predictions. We finally provide a family of "error-tolerant" auctions that maintain improved frugality ratios even if the predictions are only approximately accurate, and we provide upper bounds on their frugality ratio as a function of the prediction error.

The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm's solution and that of an optimal solution for that instance. We apply this framework to the classic k-median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed k-median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence.

Greedy k-means++ is a generalization of k-means++ where, in each iteration, a new seed is greedily chosen among multiple ℓ 2points sampled, as opposed to a single seed being sampled in k-means++. While empirical studies consistently show the superior performance of greedy k-means++, making it a preferred method in practice, a discrepancy exists between theory and practice. No theoretical justification currently explains this improved performance. Indeed, the prevailing theory suggests that greedy k-means++ exhibits worse performance than k-means++ in worst-case scenarios. This paper presents an analysis demonstrating the outperformance of the greedy algorithm compared to k-means++ for a natural class of well-separated instances with exponentially decaying distributions, such as Gaussian, specifically when ℓ = lnk +Θ(1), a common parameter setting in practical applications.

Neural Information Processing Systems http://nips.cc/

In this work we consider an online learning problem, called Online $k$-Clustering with Moving Costs, at which a learner maintains a set of $k$ facilities over $T$ rounds so as to minimize the connection cost of an adversarially selected sequence of clients. The learner is informed on the positions of the clients at each round $t$ only after its facility-selection and can use this information to update its decision in the next round. However, updating the facility positions comes with an additional moving cost based on the moving distance of the facilities. We present the first $\mathcal{O}(\log n)$-regret polynomial-time online learning algorithm guaranteeing that the overall cost (connection $+$ moving) is at most $\mathcal{O}(\log n)$ times the time-averaged connection cost of the best fixed solution. Our work improves on the recent result of (Fotakis et al., 2021) establishing $\mathcal{O}(k)$-regret guarantees only on the connection cost.

The clients' positions may change in complex and unpredictable ways and thus an a priori knowledge

Consider the following scenario: One client may get worried that the other clients may be able to obtain some information on her location by colluding and exchanging their information.

We consider three central problems in optimization: the restricted assignment load-balancing problem, the Steiner tree network design problem, and facility location clustering. We consider the online setting, where the input arrives over time, and irrevocable decisions must be made without knowledge of the future. For all these problems, any online algorithm must incur a cost that is approximately log | I | times the optimal cost in the worst-case, where | I | is the length of the input. But can we go beyond the worst-case? In this work we give algorithms that perform substantially better when a p -fraction of the input is given as a sample: the algorithm use this sample to learn a good strategy to use for the rest of the input.

In this work we consider an online learning problem, called Online k -Clustering with Moving Costs, at which a learner maintains a set of k facilities over T rounds so as to minimize the connection cost of an adversarially selected sequence of clients. The learner is informed on the positions of the clients at each round t only after its facility-selection and can use this information to update its decision in the next round. However, updating the facility positions comes with an additional moving cost based on the moving distance of the facilities. We present the first \mathcal{O}(\log n) -regret polynomial-time online learning algorithm guaranteeing that the overall cost (connection moving) is at most \mathcal{O}(\log n) times the time-averaged connection cost of the best fixed solution. Our work improves on the recent result of (Fotakis et al., 2021) establishing \mathcal{O}(k) -regret guarantees only on the connection cost.