An Improved Approximation for $k$-median, and Positive Correlation in Budgeted Optimization

TL;DR

This paper advances approximation algorithms for the $k$-median problem by developing a dependent rounding technique that balances negative and positive correlation properties, leading to improved ratio and runtime.

Contribution

It introduces a novel dependent rounding algorithm that achieves better approximation ratios and runtime dependence for the $k$-median problem compared to prior work.

Findings

Improved $k$-median approximation ratio from 2.732+ε to 2.675+ε.

Reduced runtime dependence from N^{O(1/ε^2)} to N^{O((1/ε) log(1/ε))}.

Developed a dependent rounding method balancing negative and positive correlation.

Abstract

Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to \emph{negative correlation} properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires \emph{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 $k$-median 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 $2.732 + \epsilon$ to $2.675 + \epsilon$ 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 $\epsilon$ from Li-Svensson's $N^{O(1/\epsilon^2)}$ to $N^{O((1/\epsilon) \log(1/\epsilon))}$.