Covering selfish machines

TL;DR

This paper introduces new approximation schemes and algorithms for the machine covering problem with selfish related machines, achieving near-optimal solutions efficiently and for specific cases like two machines.

Contribution

It presents the first results for selfish machines, including a monotone PTAS, an FPTAS, and improved algorithms for two machines.

Findings

Monotone PTAS with linear runtime in jobs for fixed m

FPTAS for classical machine covering problem

Improved algorithms for two-machine case

Abstract

We consider the machine covering problem for selfish related machines. For a constant number of machines, m, we show a monotone polynomial time approximation scheme (PTAS) with running time that is linear in the number of jobs. It uses a new technique for reducing the number of jobs while remaining close to the optimal solution. We also present an FPTAS for the classical machine covering problem (the previous best result was a PTAS) and use this to give a monotone FPTAS. Additionally, we give a monotone approximation algorithm with approximation ratio \min(m,(2+\eps)s_1/s_m) where \eps>0 can be chosen arbitrarily small and s_i is the (real) speed of machine i. Finally we give improved results for two machines. Our paper presents the first results for this problem in the context of selfish machines.