# Relaxing the Irrevocability Requirement for Online Graph Algorithms

## TL;DR

This paper explores models of online graph algorithms where the traditional irrevocability constraint is relaxed, allowing late acceptance or rejection of requests, leading to improved competitive ratios for certain problems.

## Contribution

It introduces new online models with relaxed irrevocability constraints and analyzes their impact on competitive ratios for various graph problems.

## Key findings

- Late Accept/Reject model achieves a constant competitive ratio for Independent Set.
- Late Accept model suffices for Vertex Cover with improved ratios.
- Matching problem's ratio improves from 2 to 3/2 in the new models.

## Abstract

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2.