## Abstract

We consider the problem of maximizing the minimum load (completion time) for machines that are controlled by selfish agents, who are only interested in maximizing their own profit. Unlike the classical load balancing problem, this problem has not been considered for selfish agents until now. The goal is to design a truthful mechanism, i.e., one in which all users have an incentive to tell the truth about the speeds of their machines. This then allows us to find good job assignments. It is known that this requires monotone approximation algorithms, in which the amount of work assigned to an agent does not increase if its bid (claimed cost per unit work) increases. 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 use an FPTAS for the classical problem, i.e., where no selfish agents are involved, to give a monotone FPTAS. Additionally, we give a monotone approximation algorithm with approximation ratio min (m, (2 + ε) s_{1} / s_{m}) where ε > 0 can be chosen arbitrarily small and s_{i} is the (real) speed of machine i. Finally we give improved results for two machines.

Original language | English |
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Pages (from-to) | 44-57 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 411 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2010 |

### Bibliographical note

Funding Information:The authors would like to thank an anonymous referee who pointed out an error in an earlier version of our approximation scheme in Section 3, another referee who helped to improve the presentation, and Motti Sorani for the helpful discussions. The second author’s research is supported by German Research Foundation (DFG). The work was performed while the author was at the University of Karlsruhe, Germany.

## Keywords

- Maximizing minimum load
- Monotone approximation algorithms
- Scheduling
- Selfish agents

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science