We consider the problem introduced by Korach and Stern (Mathematical Programming, 98: 345-414, 2003) of building a network given connectivity constraints. A network designer is given a set of vertices V and constraints S-i subset of 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 (Discrete Applied Mathematics, 156: 444-450, 2008): for the offline version of the problem, we prove an Omega(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 Omega(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 Omega(root 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 Theta(log n) on the competitive ratio for them.
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