We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data set that contains the submit...
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We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data set that contains the submitted bids of n buyers in a set of auctions, and the problem is to return personalized reserve prices r that maximize the revenue earned on these auctions by running eager second price auctions with reserve r. For this problem, which is known to be NP complete, we present a novel linear program (lp) formulation and a rounding procedure, which achieves a 0.684 approximation. This improves over the 1approximation algorithm from Rough-garden and Wang. We show that our analysis is tight for this rounding procedure. We also bound the integrality gap of the lp, which shows that it is impossible to design an algorithm that yields an approximation factor larger than 0.828 with respect to this lp.
Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for k-means in Euclidean space with...
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Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for k-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9+epsilon, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (lp) relaxation. Our techniques are quite general, and we also show improved guarantees for k-median in Euclidean metrics and for a generalization of k-means in which the underlying metric is not required to be Euclidean.
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