# Network construction with subgraph connectivity constraints

Dana Angluin, James Aspnes, and Lev Reyzin.
Network construction with subgraph connectivity constraints.
*Journal of Combinatorial Optimization*, 29(2):418–432, February 2015.

## Abstract

We consider the problem introduced by Korach and Stern (2008)
of building a network given connectivity
constraints. A network designer is given a set of vertices V and
constraints S_{i} ⊆ V, and seeks to build the lowest cost set
of edges E such that each S_{i} induces a connected subgraph of
(V,E). First, we answer a question posed by Korach and Stern:
for the offline version of the problem, we prove
an Ω(log(n)) hardness of approximation result for uniform cost
networks (where edge costs are all 1) and give an algorithm that
almost matches this bound, even in the arbitrary cost case. Then we
consider the online problem, where the constraints must be satisfied
as they arrive. We give an O(n log(n))-competitive algorithm for
the arbitrary cost online problem, which has an Ω(n)-competitive
lower bound. We look at the uniform cost case as well and give an
O(n^{2/3}log^{2/3}(n))-competitive algorithm against an oblivious
adversary, as well as an Ω(√n)-competitive lower bound
against an adaptive adversary. We also examine cases when the
underlying network graph is known to be a star or a path, and prove
matching upper and lower bounds of Θ(log(n)) on the competitive
ratio for them.

## BibTeX

Download@article{AngluinAR2015,
author = {Dana Angluin and James Aspnes and Lev Reyzin},
title = {Network construction with subgraph connectivity constraints},
month=feb,
year = 2015,
journal="Journal of Combinatorial Optimization",
volume = 29,
number = 2
}

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