Online Stochastic Matching

Barna SahaVahid Liaghat

Matching?

Adwords

Bidders𝒛𝟏 𝒛𝟐 𝒛𝟑 𝒛𝟓𝒛𝟒

𝒚𝟏𝒚𝟐 𝒚𝟑𝒚𝟏 𝒚𝟒𝒚𝟑𝒚𝟒𝒚𝟐𝒚𝟐

Adword Types: , , ,

Matching?

Adword Types

Bidders𝒛𝟏 𝒛𝟐 𝒛𝟑 𝒛𝟓𝒛𝟒

𝒏 ( 𝒚𝟏 )=𝟐𝒏 ( 𝒚𝟐 )=𝟑𝒏 ( 𝒚𝟑 )=𝟐𝒏 ( 𝒚𝟒 )=𝟐

Offline LP Relaxation

Online Matching

• Adversarial, Unknown GraphVazirani et al.[1] 1-1/e can’t do better

• Random Arrival, Unknown GraphGoel & Mehta[2] 1-1/e

can’t do better than 0.83

• i.i.d Model: Known Graph and Arrival Ratios– Integral: Bahmani et al.[3] 0.699 Can’t do better than

0.902– General: Saberi et al.[4] 0.702 Can’t do better than

0.823

𝒛𝟏 𝒛𝟐 𝒛𝟑 𝒛𝟓𝒛𝟒

𝒚𝟏𝒚𝟐 𝒚𝟑𝒚𝟏 𝒚𝟒𝒚𝟑𝒚𝟒𝒚𝟐𝒚𝟐

i.i.d. Model

𝔼 [𝑛 (𝑦 ) ]=𝑟 𝑦≤1

Competitive Ratio:

Fractional Matching

𝑓 =∑𝜔

𝐹 (𝜔 )ℙ (𝜔 )

Fractional Degree:

(Corollary 2.1 [4]) It is possible to efficiently and explicitlyconstruct (and sample from) a distribution on the set of

matchings in such that for all edges

Non-Adaptive Algorithm

Algorithm 1 - Analysis

≥0.684

Adaptive Algorithm - idea

• arrives!

• A Joint Distribution from which and are chosen.

• (i) The probability that (and ) is equal to some , is

equal to .

• (ii) Given (i), the joint the distribution is such that

the probability of is minimized.

Adaptive Algorithm - partitions

𝑓 𝑒1≥ 𝑓 𝑒2≥…≥ 𝑓 𝑒𝑘

≥ 𝑓 𝑒𝑘+1

Adaptive Algorithm

Upper Bounds

• For , no online algorithm can do better than .

• For , no online algorithm can do better than .

• For , no non-adaptive algorithm can do better than .

Questions?

References

• [1] R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An optimal algorithm for online bipartite matching. In STOC, pages 352–358. ACM, 1990.

• [2] G. Goel and A. Mehta. Online budgeted matching in random input models with applications to adwords. In SODA, pages 982–991, 2008.

• [3] B. Bahmani and M. Kapralov. Improved bounds for online stochastic matching. In ESA, pages 170–181, 2010.

• [4] V. H. Manshadi, S. Oveis Gharan, A. Saberi. Online Stochastic Matching: Online Actions Based on Offline Statistics. In SODA, 2011.