K-Medians, Facility Location, and the Chernoff-Wald Bound

TL;DR

This paper presents improved approximation algorithms for NP-hard k-medians and facility location problems, achieving better performance guarantees and introducing a new probabilistic bound called the Chernoff-Wald bound for analyzing randomized algorithms.

Contribution

It offers the first algorithms with the best known guarantees for non-metric versions of these problems and introduces the Chernoff-Wald bound for probabilistic analysis.

Findings

Improved approximation guarantee for k-medians with ln(n+n/epsilon)k medians.

Enhanced facility location solution cost bound of d+ln(n)k.

Introduction of the Chernoff-Wald bound for probabilistic analysis.

Abstract

The paper gives approximation algorithms for the k-medians and facility-location problems (both NP-hard). For k-medians, the algorithm returns a solution using at most ln(n+n/epsilon)k medians and having cost at most (1+epsilon) times the cost of the best solution that uses at most k medians. Here epsilon > 0 is an input to the algorithm. In comparison, the best previous algorithm (Jyh-Han Lin and Jeff Vitter, 1992) had a (1+1/epsilon)ln(n) term instead of the ln(n+n/epsilon) term in the performance guarantee. For facility location, the algorithm returns a solution of cost at most d+ln(n) k, provided there exists a solution of cost d+k where d is the assignment cost and k is the facility cost. In comparison, the best previous algorithm (Dorit Hochbaum, 1982) returned a solution of cost at most ln(n)(d+k). For both problems, the algorithms currently provide the best performance guarantee known for the general (non-metric) problems. The paper also introduces a new probabilistic bound (called "Chernoff-Wald bound") for bounding the expectation of the maximum of a collection of sums of random variables, when each sum contains a random number of terms. The bound is used to analyze the randomized rounding scheme that underlies the algorithms.