Thresholded Covering Algorithms for Robust and Max-Min Optimization

TL;DR

This paper introduces a simple threshold-based template for k-robust covering problems, leading to improved approximation algorithms for several classic optimization problems under uncertainty.

Contribution

The paper presents a new, intuitive template for k-robust problems that yields improved approximation algorithms for Steiner tree, set cover, Steiner forest, minimum-cut, and multicut.

Findings

Improved approximation algorithms for k-robust Steiner tree and set cover.

First approximation algorithms for k-robust Steiner forest, minimum-cut, and multicut.

Algorithms for max-min problems identifying the most costly elements to cover.

Abstract

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".