Robust Online Algorithms for Dynamic Choosing Problems

TL;DR

This paper introduces a framework for semi-online algorithms that maximize profit in dynamic choosing problems, achieving near-optimal competitiveness with bounded migration, applicable to problems like Knapsack and Maximum Independent Set.

Contribution

It develops a method to convert offline approximation algorithms into semi-online algorithms with tight bounds on migration for maximizing profit in choosing problems.

Findings

Achieves $( ext{approximation factor}- ext{epsilon})$-competitiveness.

Provides lower bounds showing no better migration than $O(1/ extepsilon)$ is possible.

Applies to a broad class of problems including Knapsack and Maximum Independent Set.

Abstract

Semi-online algorithms that are allowed to perform a bounded amount of repacking achieve guaranteed good worst-case behaviour in a more realistic setting. Most of the previous works focused on minimization problems that aim to minimize some costs. In this work, we study maximization problems that aim to maximize their profit. We mostly focus on a class of problems that we call choosing problems, where a maximum profit subset of a set objects has to be maintained. Many known problems, such as Knapsack, MaximumIndependentSet and variations of these, are part of this class. We present a framework for choosing problems that allows us to transfer offline $\alpha$-approximation algorithms into $(\alpha-epsilon)$-competitive semi-online algorithms with amortized migration $O(1/\epsilon)$. Moreover we complement these positive results with lower bounds that show that our results are tight in the sense that no amortized migration of $o(1/\epsilon)$ is possible.