# Online Constrained Forest and Prize-Collecting Network Design

## TL;DR

This paper introduces an online algorithm for a broad class of network design problems, achieving logarithmic competitiveness and extending to cases with penalty-based constraint violations.

## Contribution

It generalizes previous algorithms for online network design and penalty-allowing variants, providing a unified O(log k)-competitive solution.

## Key findings

- Developed an O(log k)-competitive online algorithm for general network design.
- Extended the algorithm to handle penalties for connectivity violations.
- Unified previous approaches into a broader, more versatile framework.

## Abstract

In this paper, we study a very general type of online network design problem, and generalize two different previous algorithms, one for an online network design problem due to Berman and Coulston [4] and one for (offline) general network design problems due to Goemans and Williamson [9]; we give an O(log k)-competitive algorithm, where k is the number of nodes that must be connected. We also consider a further generalization of the problem that allows us to pay penalties in exchange for violating connectivity constraints; we give an online O(log k)-competitive algorithm for this case as well.