Buyback Problem - Approximate matroid intersection with cancellation costs

TL;DR

This paper introduces a deterministic algorithm for the buyback problem under matroid intersection constraints, providing optimal competitive ratios and extending to general downward closed set systems, with applications in various resource allocation scenarios.

Contribution

It extends previous work by developing a deterministic algorithm for buyback problems with intersecting matroid constraints and establishes a matching lower bound on the competitive ratio.

Findings

Developed a deterministic algorithm for buyback with matroid intersection constraints.

Proved a matching lower bound on the competitive ratio.

Extended results to arbitrary downward closed set systems.

Abstract

In the buyback problem, an algorithm observes a sequence of bids and must decide whether to accept each bid at the moment it arrives, subject to some constraints on the set of accepted bids. Decisions to reject bids are irrevocable, whereas decisions to accept bids may be canceled at a cost that is a fixed fraction of the bid value. Previous to our work, deterministic and randomized algorithms were known when the constraint is a matroid constraint. We extend this and give a deterministic algorithm for the case when the constraint is an intersection of $k$ matroid constraints. We further prove a matching lower bound on the competitive ratio for this problem and extend our results to arbitrary downward closed set systems. This problem has applications to banner advertisement, semi-streaming, routing, load balancing and other problems where preemption or cancellation of previous allocations is allowed.