Dana Angluin, James Aspnes, and Lev Reyzin. Network construction with subgraph connectivity constraints. Journal of Combinatorial Optimization, 29(2):418–432, February 2015.
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 Si ⊆ V, and seeks to build the lowest cost set of edges E such that each Si 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(n2/3log2/3(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.
@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
}