# 1 Local Computation Mechanism Design Shai Vardi, Tel Aviv University Joint work with Avinatan Hassidim & Yishay Mansour UC Berekely, February 18 th, 2014.

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1 Local Computation Mechanism Design Shai Vardi, Tel Aviv University Joint work with Avinatan Hassidim & Yishay Mansour UC Berekely, February 18 th, 2014

2 Motivation  Imagine a huge auction, with millions of items and hundreds of thousands of buyers. An item arrives to be shipped. We don’t want to have to compute the result of the entire auction, just to know to which buyer to ship the item….  There is a cloud with millions of computers on which we would like to schedule millions of jobs. We are queried on a job and would like to reply on which machine it should run. But we don’t want to compute the entire schedule….

Example – a combinatorial auction

Which item does Dudu get?

When do we need Local Computation Algorithms (LCAs)?  Huge input.  Not enough time or space to calculate the entire solution.  Only need small parts of the solution at any one time. Local Computation Algorithms 5

6 LCAs implement query access to a global solution. We require LCAs to be:  Fast – at most polylogarithmic in the input size per query.  Space-efficient – at most polylogarithmic overall.  Replies to all queries are consistent with the same solution.

7 Related work  Local computation of PageRank contributions Andersen et al., 2008.  Property-preserving data reconstruction Ailon et al., 2008.  On the efficiency of local decoding procedures for error- correcting codes Katz and Trevisan, 2000.

8 Previous work on LCAs  Fast local computation algorithms Rubinfeld, Tamir, V and Xie, 2011.  Space-efficient local computation algorithms Alon, Rubinfeld, V and Xie, 2012.  Converting online algorithms to local computation algorithms Mansour, Rubinstein, V and Xie, 2012.  Tighter bounds for local computations Reingold and V, 2014.

Local computation mechanism design

Mechanisms without payment Usually, the allocation is harder to find than the payments. This is especially true considering the large volume of work on calculating payments in the past few years, e.g., Babaioff, Kleinberg, Silvkins, EC ‘10, EC ‘13. Therefore, I won’t get into payments in this talk. I will focus on mechanisms without payments.

Stable matching The problem: There is a group of men and women. Each man has a preference list over the women, and each woman has a preference list over the men. A stable matching is one in which there is no man and woman who both prefer each other over their matched partner.

Stable matching The Gale-Shapley algorithm (’62) (male courtship): Each man goes to his most preferred woman. Each woman chooses which man to keep. The rejected men go to their next-preferred woman. Each woman once again chooses which man to keep, and so on.

Stable matching

14 Restrictions

Restrictions for stable matching 1 2 1 2 2 1 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 1 2 2 1 1 22 1 1 1 1 2 1

Local mechanism for stable matching

Previous work  Truncation strategies in matching markets. Roth and Rothblum,1999.  A parallel iterative improvement stable matching algorithm. Lu and Zheng, 2003.  Marriage, honesty, and stability. Immorlica and Mahdian, 2005.  Almost stable matchings by truncating the Gale-Shapley algorithm. Floreen et al., 2010.

Stable matching LCA

Proof intuition – why is it local?

Stable matching LCA

How many men are left unmatched? k=1

k=2

Stable matching LCA

Proof of 3)

From monotonicity

Stable matching LCA recap

Some more local computation mechanisms

The non-local case We show how to apply our techniques to the non-local case.

Approximating the Gale-Shapley algorithm

2-approximating the Gale-Shapley algorithm

Part 1 of the proof

Part 2 of the proof

Gale-Shapley approximation recap

44 Thank you for listening! Questions?

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