Online Generalized Network Design Under (Dis)Economies of Scale
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We consider a general online network design problem where a sequence of N requests arrive over time, each of which needs to use some subset of the available resources E. The cost incurred by any resource e is some function f_e of the total load L_e on that resource. The objective is to minimize the total cost ∑_e∈ E f_e(L_e). We focus on cost functions that exhibit (dis)economies of scale, that are of the form f_e(x) = σ_e + ξ_e· x^α_e if x>0 (and zero if x=0), where the exponent α_e≥ 1. Optimization problems under these functions have received significant recent attention due to applications in energy-efficient computing. Our main result is a deterministic online algorithm with tight competitive ratio Θ(max_e∈ E(σ_e/ξ_e)^1/α_e) when α_e is constant for all e∈ E. This framework is applicable to a variety of network design problems in undirected and directed graphs, including multicommodity routing, Steiner tree/forest connectivity and set-connectivity. In fact, our online competitive ratio even matches the previous-best (offline) approximation ratio for generalized network design.
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