### Abstract

We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain these approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith's Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic non-preemptive strongly local policies. Since Smith's Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed. We prove that ProportionalSharing, a preemptive strongly local policy, achieves an approximation ratio of 2.618 for the weighted sum of completion times, and an approximation ratio of 2.5 in the unweighted case. We also observe that these bounds are tight. Next, we consider Rand, a natural non-preemptive but randomized policy. We show that it achieves an approximation ratio of at most 2.13; moreover, if the sum of the weighted completion times is negligible compared to the cost of the optimal solution, this improves to π/2. Finally, we show that both ProportionalSharing and Rand induce potential games, and thus always have a pure Nash equilibrium (unlike Smith's Rule). This allows us to design the first combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling. It achieves a factor of 2+∈ for any ∈ > 0, and involves imitating best response dynamics using a variant of ProportionalSharing as the policy.

Original language | English (US) |
---|---|

Title of host publication | STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing |

Pages | 539-548 |

Number of pages | 10 |

DOIs | |

State | Published - 2011 |

Event | 43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States Duration: Jun 6 2011 → Jun 8 2011 |

### Other

Other | 43rd ACM Symposium on Theory of Computing, STOC'11 |
---|---|

Country | United States |

City | San Jose, CA |

Period | 6/6/11 → 6/8/11 |

### Fingerprint

### Keywords

- approximation algorithm
- coordination mechanisms
- price of anarchy
- scheduling

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing*(pp. 539-548) https://doi.org/10.1145/1993636.1993708

**Inner product spaces for MinSum coordination mechanisms.** / Cole, Richard; Correa, José R.; Gkatzelis, Vasilis; Mirrokni, Vahab; Olver, Neil.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing.*pp. 539-548, 43rd ACM Symposium on Theory of Computing, STOC'11, San Jose, CA, United States, 6/6/11. https://doi.org/10.1145/1993636.1993708

}

TY - GEN

T1 - Inner product spaces for MinSum coordination mechanisms

AU - Cole, Richard

AU - Correa, José R.

AU - Gkatzelis, Vasilis

AU - Mirrokni, Vahab

AU - Olver, Neil

PY - 2011

Y1 - 2011

N2 - We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain these approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith's Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic non-preemptive strongly local policies. Since Smith's Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed. We prove that ProportionalSharing, a preemptive strongly local policy, achieves an approximation ratio of 2.618 for the weighted sum of completion times, and an approximation ratio of 2.5 in the unweighted case. We also observe that these bounds are tight. Next, we consider Rand, a natural non-preemptive but randomized policy. We show that it achieves an approximation ratio of at most 2.13; moreover, if the sum of the weighted completion times is negligible compared to the cost of the optimal solution, this improves to π/2. Finally, we show that both ProportionalSharing and Rand induce potential games, and thus always have a pure Nash equilibrium (unlike Smith's Rule). This allows us to design the first combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling. It achieves a factor of 2+∈ for any ∈ > 0, and involves imitating best response dynamics using a variant of ProportionalSharing as the policy.

AB - We study coordination mechanisms aiming to minimize the weighted sum of completion times of jobs in the context of selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain these approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith's Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic non-preemptive strongly local policies. Since Smith's Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed. We prove that ProportionalSharing, a preemptive strongly local policy, achieves an approximation ratio of 2.618 for the weighted sum of completion times, and an approximation ratio of 2.5 in the unweighted case. We also observe that these bounds are tight. Next, we consider Rand, a natural non-preemptive but randomized policy. We show that it achieves an approximation ratio of at most 2.13; moreover, if the sum of the weighted completion times is negligible compared to the cost of the optimal solution, this improves to π/2. Finally, we show that both ProportionalSharing and Rand induce potential games, and thus always have a pure Nash equilibrium (unlike Smith's Rule). This allows us to design the first combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling. It achieves a factor of 2+∈ for any ∈ > 0, and involves imitating best response dynamics using a variant of ProportionalSharing as the policy.

KW - approximation algorithm

KW - coordination mechanisms

KW - price of anarchy

KW - scheduling

UR - http://www.scopus.com/inward/record.url?scp=79959694892&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959694892&partnerID=8YFLogxK

U2 - 10.1145/1993636.1993708

DO - 10.1145/1993636.1993708

M3 - Conference contribution

SN - 9781450306911

SP - 539

EP - 548

BT - STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing

ER -