Online Generalized Network Design Under (Dis)Economies of Scale

TL;DR

This paper introduces a deterministic online algorithm for a broad class of network design problems with (dis)economies of scale, achieving a competitive ratio that matches offline approximations and is applicable to various network optimization tasks.

Contribution

The paper presents a new online algorithm with tight competitive ratio for generalized network design problems with (dis)economies of scale, covering multiple network optimization scenarios.

Findings

Achieves tight competitive ratio depending on resource parameters.

Applicable to multiple network design problems including routing and connectivity.

Matches previous offline approximation ratios in an online setting.

Abstract

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 $\sum_{e\in E} f_e(L_e)$. We focus on cost functions that exhibit (dis)economies of scale, that are of the form $f_e(x) = \sigma_e + \xi_e\cdot x^{\alpha_e}$ if $x>0$ (and zero if $x=0$), where the exponent $\alpha_e\ge 1$. 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 $\Theta\left(\max_{e\in E} \left(\frac{\sigma_e}{\xi_e}\right)^{1/\alpha_e}\right)$ when $\alpha_e$ is constant for all $e\in E$. 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.