# Network Construction with Ordered Constraints

## TL;DR

This paper investigates constructing networks from ordered connectivity constraints, providing complexity results and optimal algorithms for specific graph classes, advancing understanding of online and offline network design problems.

## Contribution

It introduces the concept of ordered constraints in network construction, establishes hardness results, and develops optimal algorithms for certain graph subclasses.

## Key findings

- Hardness of approximation results for offline problems
- Exponential improvements in online algorithms for general graphs
- Optimal competitive algorithms for star and path graph classes

## Abstract

In this paper, we study the problem of constructing a network by observing ordered connectivity constraints, which we define herein. These ordered constraints are made to capture realistic properties of real-world problems that are not reflected in previous, more general models. We give hardness of approximation results and nearly-matching upper bounds for the offline problem, and we study the online problem in both general graphs and restricted sub-classes. In the online problem, for general graphs, we give exponentially better upper bounds than exist for algorithms for general connectivity problems. For the restricted classes of stars and paths we are able to find algorithms with optimal competitive ratios, the latter of which involve analysis using a potential function defined over pq-trees.