Randomized Online Algorithms for the Buyback Problem

TL;DR

This paper introduces a new randomized algorithm for the matroid buyback problem, establishing optimal competitive ratios against oblivious adversaries and proving limitations against adaptive adversaries, thus fully characterizing achievable performance.

Contribution

The paper presents a novel randomized algorithm with proven optimal competitive ratio for the matroid buyback problem, resolving the limits of randomized approaches under different adversary models.

Findings

The algorithm achieves the best possible competitive ratio against oblivious adversaries.

No randomized algorithm can outperform the optimal deterministic algorithm against adaptive adversaries.

The work fully characterizes the competitive ratios achievable by randomized algorithms for this problem.

Abstract

In the matroid buyback problem, an algorithm observes a sequence of bids and must decide whether to accept each bid at the moment it arrives, subject to a matroid constraint on the set of accepted bids. Decisions to reject bids are irrevocable, whereas decisions to accept bids may be canceled at a cost which is a fixed fraction of the bid value. We present a new randomized algorithm for this problem, and we prove matching upper and lower bounds to establish that the competitive ratio of this algorithm, against an oblivious adversary, is the best possible. We also observe that when the adversary is adaptive, no randomized algorithm can improve the competitive ratio of the optimal deterministic algorithm. Thus, our work completely resolves the question of what competitive ratios can be achieved by randomized algorithms for the matroid buyback problem.