online stochastic matching barna saha vahid liaghat
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Online Stochastic Matching
Barna SahaVahid Liaghat
Matching?
Adwords
Bidders๐๐ ๐๐ ๐๐ ๐๐๐๐
๐๐๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐
Adword Types: , , ,
Matching?
Adword Types
Bidders๐๐ ๐๐ ๐๐ ๐๐๐๐
๐ ( ๐๐ )=๐๐ ( ๐๐ )=๐๐ ( ๐๐ )=๐๐ ( ๐๐ )=๐
Offline LP Relaxation
Online Matching
โข Adversarial, Unknown GraphVazirani et al.[1] 1-1/e canโt do better
โข Random Arrival, Unknown GraphGoel & Mehta[2] 1-1/e
canโt do better than 0.83
โข i.i.d Model: Known Graph and Arrival Ratiosโ Integral: Bahmani et al.[3] 0.699 Canโt do better than
0.902โ General: Saberi et al.[4] 0.702 Canโt do better than
0.823
๐๐ ๐๐ ๐๐ ๐๐๐๐
๐๐๐๐ ๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐
i.i.d. Model
๐ผ [๐ (๐ฆ ) ]=๐ ๐ฆโค1
Competitive Ratio:
Fractional Matching
๐ =โ๐
๐น (๐ )โ (๐ )
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
โฅ0.684
Adaptive Algorithm - idea
โข arrives!
โข A Joint Distribution from which and are chosen.
โข (i) The probability that (and ) is equal to some , is
equal to .
โข (ii) Given (i), the joint the distribution is such that
the probability of is minimized.
Adaptive Algorithm - partitions
๐ ๐1โฅ ๐ ๐2โฅโฆโฅ ๐ ๐๐
โฅ ๐ ๐๐+1
Adaptive Algorithm
Upper Bounds
โข For , no online algorithm can do better than .
โข For , no online algorithm can do better than .
โข For , no non-adaptive algorithm can do better than .
Questions?
References
โข [1] R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An optimal algorithm for online bipartite matching. In STOC, pages 352โ358. ACM, 1990.
โข [2] G. Goel and A. Mehta. Online budgeted matching in random input models with applications to adwords. In SODA, pages 982โ991, 2008.
โข [3] B. Bahmani and M. Kapralov. Improved bounds for online stochastic matching. In ESA, pages 170โ181, 2010.
โข [4] V. H. Manshadi, S. Oveis Gharan, A. Saberi. Online Stochastic Matching: Online Actions Based on Offline Statistics. In SODA, 2011.