Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency

TL;DR

This paper develops a versatile, fully combinatorial approximation framework for generalized network design problems with (dis)economies of scale cost functions, applicable to energy efficiency and unrelated machine scheduling, with ratios independent of problem size.

Contribution

It introduces the first non-trivial approximation algorithm for GND with (D)oS costs that is fully combinatorial, generalizes to various request types, and is independent of problem size.

Findings

Provides a generic approximation framework for (D)oS cost functions.

First approximation for scheduling unrelated machines with (D)oS costs.

Framework is strongly polynomial and based on smoothness techniques.

Abstract

In a generalized network design (GND) problem, a set of resources are assigned to multiple communication requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource specific cost function. For example, a request may be to establish a virtual circuit, thus contributing to the load on each edge in the circuit. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current paper advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) we present a generic approximation framework that yields approximation results for a much wider family of requests in both directed and undirected graphs; (2) our framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) our framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current paper is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources' technology and does not grow with the number of resources, the number of requests, or their weights. The design of our framework relies heavily on Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.