An Improved Approximation for k -Median and Positive Correlation in Budgeted Optimization

Abstract

Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to negative correlation properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires positive correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly best-possible behavior -near-independence, which generalizes positive correlation -on "small" subsets of the variables. The recent breakthrough of Li & Svensson for the classical kmedian problem has to handle positive correlation in certain dependent-rounding settings, and does so implicitly. We improve upon Li-Svensson's approximation ratio for k-median from 2.732+ to 2.675+ by developing an algorithm that improves upon various aspects of their work. Our dependent-rounding approach helps us improve the dependence of the runtime on the parameter from Li-Svensson's N O(1/ 2 ) to N O((1/ ) log(1/ )) .

Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms

“…Indeed, all recent progress on the approximation of k-median beyond long-standing local search approaches [4] has involved reducing the factor 2 that is lost by Jain and Vazirani when two solutions are combined to open exactly k facilities (i.e. in the rounding of a so-called bipoint solution) [22,9]. Here, we show that it is possible to reduce this loss all the way to (1 + ǫ) by fundamentally changing the way in which dual solutions are constructed and maintained.…”

“…Here we are given a set D of n points in R ℓ and a set F of m points in R ℓ corresponding to facilities. The task is to select a set S of at most k facilities from F so as to minimize j∈D c(j, S), where c(j, S) is now the (non-squared) Euclidean distance from j to its nearest facility in S. For this problem, no approximation better than the general 2.675-approximation algorithm of Byrka et al [9] for k-median was known. In the second extension, we consider a variant of the k-means problem in which each c(j, S) corresponds to the squared distance in an arbitrary (possibly non-Euclidean) metric on D ∪ F. For this problem, the best-known approximation algorithm is a 16-approximation due to Gupta and Tangwongsan [15].…”

“…9 Note that each value α (0) j is fixed at the beginning of RaisePrice, and there are only a (relatively) small number of choices for σ such that any given j is in W(σ). Thus, if we choose σ uniformly at random, the probability that any given j ∈ W(σ) is small.…”

“…Over the past several decades, a toolbox of core algorithmic techniques such as dual fitting, primal-dual and LP-rounding, has been refined and applied to these problems leading to improved approximation guarantees [27,11,8,21,23,18,17,22,16]. In particular, the current best approximation guarantees for both facility location (a 1.488-approximation due to Li [21]) and k-median (a 2.675-approximation due to Byrka et al [9]) are LP-based and give significantly better results than previous local search algorithms [4,10]. However, such LP-based techniques have not yet been able to attain similar improvements for k-means.…”

An Improved Approximation Algorithm for Knapsack Median Using Sparsification

“…k-median was first approximated within constant factor 6 2 3 in 1999 [2], with a series of improvements leading to the current best-known factor of 2.674 [1] 1 . KM was first studied in 2011 by Krishnaswamy et.…”