Local Computation Mechanism Design

1. Local Computation Mechanism Design ShaiVardi, Tel Aviv University Joint work with Avinatan Hassidim & Yishay Mansour UC Berekely, February 18th, 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….

3. Example – a combinatorial auction

4. Which item does Dudu get?

5. Local Computation Algorithms • 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.

6. Local Computation Algorithms • 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.

9. Local computation mechanism design • We want to implement a local mechanism • A local allocation algorithm . • A local payment scheme . • The replies of and to all the queries combine to a “good” allocation and payment scheme that guarantees truthfulness. • The truthfulness must be guaranteed regardless of other queries or the locality of the mechanism.

10. 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.

11. 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.

12. 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.

13. Stable matching

14. Restrictions It is clear that a problem that can not be solved polynomially does not have an LCA. Still, not every problem that can be solved polynomially necessarily has an LCA. For example, it does not seem likely there is an LCA for maximal independent set on a graph where many nodes have degree We will usually have some restrictions on our game space, e.g., constant degree, randomness etc.

15. Restrictions for stable matching 2 1 1 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 2 2 2 2 2 1 2 1 2 1 2 1

16. Local mechanism for stable matching • Assume that the men’s preferences are chosen uniformly i.i.d. • We allow men to be unmatched; we also allow the mechanism to “disqualify” men: they are unmatched but unable to contest the matching • (they are sometimes referred to as “unstable” or “blocking”, and such matchings are said to be “almost stable”.) • Our goal: a stable matching, with at most unmatched men and at most disqualified men.

17. 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.

18. Stable matching LCA • We propose the following local computation algorithm, for , : • Allow the men’s lists to be of length • ). • Simulate the Gale-Shapley men’s courtship algorithm for) rounds. • The men rejected on the last round are disqualified.

19. Stable matching LCA We need to prove 3 things: The algorithm is an LCA (i.e., runs in polylog time and space). At mostof the men will remain unmatched because the list length is bounded by . At most of the men will be disqualified because they were rejected on the last round.

20. Proof intuition – why is it local? Assume that the women’s lists are of length as well.

21. Proof intuition – why is it local? If the degrees were all bounded by and we run Gale-Shapley for rounds, the maximum number of queries would be . However, the men’s degree is and the women’s degree is distributed binomially We show that in the worst case (w.h.p.), we need queries.

22. Stable matching LCA We need to prove 3 things: The algorithm is an LCA (i.e., runs in polylog time and space). At mostof the men will remain unmatched because the list length is bounded by . At most of the men will be disqualified because they were rejected on the last round.

23. How many men are left unmatched? k=1 Worst case: Average case: With prob. :

24. k=2 Worst case: Average case: With prob. :

25. Stable matching LCA We need to prove 3 things: The algorithm is an LCA (i.e., runs in polylog time and space). At mostof the men will remain unmatched because the list length is bounded by . At most of the men will be disqualified because they were rejected on the last round.

26. Proof of 3) Denote by the number of men rejected on round Observation: is monotonically non-increasing in (Whenever a woman is matched she remains matched).

27. Proof of 3) Claim: The number of men rejected on round is at most Proof: The total number of rejections possible is From monotonicity

28. Stable matching LCA recap • We propose the following local computation algorithm, for , : • Allow the men’s lists to be of length • ). • Simulate the Gale-Shapley men’s courtship algorithm for) rounds. • The men rejected on the last round are disqualified.

29. Some more local computation mechanisms • - approximation universally truthful local mechanism for - single minded bidders. • - approximation truthful-in-expectation local mechanism for machine scheduling on related machines. • An LCA which simulates Random Serial Dictatorship. • Some others similar results.

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

31. Approximating the Gale-Shapley algorithm Assume that the Gale-Shapley algorithm produces the matching . We would like to show that after a constant number of iterations, we get a matching that is almost the size of . Note: we don’t assume uniform choice henceforth, only that the men’s lists are of bounded length.

32. Approximating the Gale-Shapley algorithm We already have an additive bound: We saw that the number of men rejected on round is at most Therefore, after rounds, we will have a matching of size at least We would like a multiplicative bound too.

33. Approximating the Gale-Shapley algorithm In the paper, we show an approximation. Today, we will show a 2-approximation.

34. 2-approximating the Gale-Shapley algorithm Notation: in round - the size of the matching in round - the total number of men who have been rejected by all of their women . - the number of men rejected in round .

35. 2-approximating the Gale-Shapley algorithm

36. 2-approximating the Gale-Shapley algorithm Notice that . Therefore, , it holds that and we are done. Hence we can assume

37. 2-approximating the Gale-Shapley algorithm is monotonically increasing, so for - the number of men rejected in round (monotonically decreasing).

38. 2-approximating the Gale-Shapley algorithm The proof is in two parts: After rounds, there are few possible total rejections left. Thereafter, the number of rejections left are reduced by a constant in each round. Thus, after a constant number of rounds, there are fewer than total rejections left.

39. Part 1 of the proof for

40. Part 2 of the proof - the number of remaining possible rejections. Because ,

41. Part 2 of the proof Because , and so

42. Part 2 of the proof Because We take ,

43. Gale-Shapley approximation recap Theorem: Consider a stable matching problem. Let each man's list be bounded by . Denote the size of the stable matching returned by the Gale-Shapley algorithm by Then, if the process is stopped after rounds, the matching returned is at most a -approximation to , and has at most unstable couples.

44. Thank you for listening!Questions?