Minimum-Cost Network Design with (Dis)economies of Scale

Given a network, a set of demands, and a cost function $f(\cdot)$, the min-cost network design problem is to route all demands with the objective of minimizing $\sum_e f(\ell_e)$, where $\ell_e$ is the total traffic load under the routing. We focus on cost functions of the form $f(x)= \sigma + x^{\alpha}$ for $x > 0$, with $f(0) = 0$. For $\alpha \le 1$, $f(\cdot)$ is subadditive. This case 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 $\alpha > 1$ with a positive start-up cost $\sigma > 0$. Now, the cost function $f(\cdot)$ is neither subadditive nor superadditive. It aims to model a range of computing and communication devices for which doubling processing speed more than doubles their power consumption. 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 contribut...