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Online Stochastic Matching
Barna SahaVahid Liaghat
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Matching?
Adwords
Biddersππ ππ ππ ππππ
ππππ ππππ ππππππππππ
Adword Types: , , ,
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Matching?
Adword Types
Biddersππ ππ ππ ππππ
π ( ππ )=ππ ( ππ )=ππ ( ππ )=ππ ( ππ )=π
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Offline LP Relaxation
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Online Matching
β’ Adversarial, Unknown GraphVazirani et al.[1] 1-1/e canβt do better
β’ Random Arrival, Unknown GraphGoel & Mehta[2] 1-1/e
canβt do better than 0.83
β’ i.i.d Model: Known Graph and Arrival Ratiosβ Integral: Bahmani et al.[3] 0.699 Canβt do better than
0.902β General: Saberi et al.[4] 0.702 Canβt do better than
0.823
ππ ππ ππ ππππ
ππππ ππππ ππππππππππ
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i.i.d. Model
πΌ [π (π¦ ) ]=π π¦β€1
Competitive Ratio:
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Fractional Matching
π =βπ
πΉ (π )β (π )
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
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Non-Adaptive Algorithm
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Algorithm 1 - Analysis
β₯0.684
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Adaptive Algorithm - idea
β’ arrives!
β’ A Joint Distribution from which and are chosen.
β’ (i) The probability that (and ) is equal to some , is
equal to .
β’ (ii) Given (i), the joint the distribution is such that
the probability of is minimized.
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Adaptive Algorithm - partitions
π π1β₯ π π2β₯β¦β₯ π ππ
β₯ π ππ+1
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Adaptive Algorithm
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Upper Bounds
β’ For , no online algorithm can do better than .
β’ For , no online algorithm can do better than .
β’ For , no non-adaptive algorithm can do better than .
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Questions?
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References
β’ [1] R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An optimal algorithm for online bipartite matching. In STOC, pages 352β358. ACM, 1990.
β’ [2] G. Goel and A. Mehta. Online budgeted matching in random input models with applications to adwords. In SODA, pages 982β991, 2008.
β’ [3] B. Bahmani and M. Kapralov. Improved bounds for online stochastic matching. In ESA, pages 170β181, 2010.
β’ [4] V. H. Manshadi, S. Oveis Gharan, A. Saberi. Online Stochastic Matching: Online Actions Based on Offline Statistics. In SODA, 2011.