Given a network, a set of demands and a cost function f (·), the min-cost network design problem is to route all demands with the objective of minimizing e f ( e), where e is the total traffic load under the routing. We focus on cost functions of the form f (x) = σ+x α for x > 0, with f (0) = 0. For α ≤ 1, f (·) is subadditive and exhibits behavior consistent with economies of scale. This problem corresponds to the well-studied Buy-at-Bulk network design problem and admits polylogarithmic approximation and hardness.In this paper, we focus on the less studied scenario of α > 1 with a positive startup cost σ > 0. Now, the cost function f (·) is neither subadditive nor superadditive. This is motivated by minimizing network-wide energy consumption when supporting a set of traffic demands. It is commonly accepted that, for some computing and communication devices, doubling processing speed more than doubles the energy consumption. Hence, in Economics parlance, such a cost function reflects diseconomies of scale.We begin by discussing why existing routing techniques such as randomized rounding and tree-metric embedding fail to generalize directly. We then present our main contribution, which is a polylogarithmic approximation algorithm. We obtain this result by first deriving a bicriteria approximation for a related capacitated min-cost flow problem that we believe is interesting in its own right. Our approach for this problem builds upon the well-linked decomposition due to Chekuri-Khanna-Shepherd [1], the construction of expanders via matchings due to , and edge-disjoint routing in well-connected graphs due to . However, we also develop new techniques that allow us to keep a handle on the total cost, which was not a concern in the aforementioned literature.
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