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Computational Advertising Bipartite Graph Matching
13CSE001
Online Algorithms
• Classic model of algorithms – We get to see the entire input, then compute some
function of it
– In this context, “offline algorithm”
• Online Algorithms – We get to see the input one piece at a time, and need
to make irrevocable decisions along the way
– Similar to the data stream model
Example: Bipartite Matching
Nodes: Boys and Girls; Edges: Compatible Pairs Goal: Match as many compatible pairs as possible
Example: Bipartite Matching
Example: Bipartite Matching
Perfect matching... all the vertices of graph are matched Maximum Matching... a matching that contains the longest possible number of matches
Matching Algorithm
• Problem: Find a maximum matching for a given bipartite graph
– A perfect one if it exists
• There is a polynomial – time offline algorithm based on augmenting path (Hopcroft & Karp 1973)
Online Graph Matching Problem
• Initially, we are given the sets boys and girls
• In each round, one girl’s choice are revealed
– That is, girl’s edges are revealed
• At that time, we have to decide either:
– Pair the girl with a boy
– Do not pair the girl with any boy
• Example of application:
Assigning tasks to servers
Online Graph Matching: Example
Online Graph Matching: Example
Online Graph Matching: Example
Online Graph Matching: Example
Online Graph Matching: Example
Online Graph Matching: Example
Online Graph Matching: Example
Online Graph Matching: Example
Greedy Algorithm
• Greedy algorithm for the online graph matching problem:
– Pair the new girl with any eligible boy
• If there is none, do not pair girl
• How good is the algorithm?
Competitive Ratio
• For input I, suppose greedy produces matching 𝑴𝒈𝒓𝒆𝒆𝒅𝒚 while an optimal matching
is 𝑴𝒐𝒑𝒕
Competitive ratio=
𝒎𝒊𝒏𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑖𝑛𝑝𝑢𝑡𝑠 𝐼(| 𝑴𝒈𝒓𝒆𝒆𝒅𝒚|/ |𝑴𝒐𝒑𝒕|)
Analyzing the Greedy Algorithm • Suppose 𝑴𝒈𝒓𝒆𝒆𝒅𝒚 ≠ 𝑴𝒐𝒑𝒕
• Consider the set G of girls
matched 𝑴𝒐𝒑𝒕 but not in 𝑴𝒈𝒓𝒆𝒆𝒅𝒚
• (1) |𝑴𝒐𝒑𝒕| ≤ |𝑴𝒈𝒓𝒆𝒆𝒅𝒚|+|G|
• Every boy B adjacent to girls in G
is already matched in 𝑴𝒈𝒓𝒆𝒆𝒅𝒚
• (2) |𝑴𝒈𝒓𝒆𝒆𝒅𝒚|≥ |B|
• Optimal matches all the girls in G to boys in B
• (3) |G|≤|B|
Analyzing the Greedy Algorithm
Combining (2) and (3):
• (4) |G|≤|B|≤ |𝑴𝒈𝒓𝒆𝒆𝒅𝒚|
Combining (1) and (4):
|𝑴𝒐𝒑𝒕| ≤ |𝑴𝒈𝒓𝒆𝒆𝒅𝒚|+|𝑴𝒈𝒓𝒆𝒆𝒅𝒚|
|𝑴𝒐𝒑𝒕| ≤ 2|𝑴𝒈𝒓𝒆𝒆𝒅𝒚|
|𝑴𝒈𝒓𝒆𝒆𝒅𝒚|/ |𝑴𝒐𝒑𝒕|≥1/2
History of Web Advertising
• Banner ads (1995-2001)
– Initial form of web advertising
– Popular websites charged X$
for every 1000 “impressions”
of the add
• Called “CPM” rate (cost per thousand impressions)
• Modeled similar to TV, magazine ads
– From untargeted to demographically targeted
– Low click-through rates
• Low ROI for advertisers
Performance-based Advertising
• Introduced by Overture around 2000
– Advertisers bid on search keywords
– When someone searches for that keyword, the highest bidder’s ad is shown
– Advertiser is charged only if the ad is clicked on
• Similar model adopted by Google with some changes around 2002
– Called Adwords
Algorithmic Challenges
• Performance – based advertising works – Multi-billion-dollar industry
• What ads to show for a given query? – The AdWords Problem Mining of Massive Datasets
• If I am an advertiser, which search terms should I bid on and how much should I bid? – (It’s not our focus)
AdWords Problem
• A stream of queries arrives at the search engine: 𝑞1, 𝑞2, …
• Several advertisers bid on each query
• When query 𝑞𝑖 arrives, search engine must pick a subset of advertisers whose ads are shown
• Goal: Maximize search engine’s revenues
• Clearly we need an online algorithm!
Expected Revenue
Expected Revenue
The AdWords Innovation
Instead of sorting advertisers by bid, sort by expected revenue
Limitations of Simple Algorithm
• CTR of an ad is unknown
• Advertisers have limited budgets and bid on multiple ads.
Future Work
• Estimation of CTR
• Algorithm to solve limited budgets problem
(Balance Algorithm)
References
• [1] Mehta A, Saberi A, Vazirani U, Vazirani V. Adwords and generalized online matching. J ACM (JACM) 2007;54(5):22.
• [2] Reiss C, Wilkes J, Hellerstein JL. Google cluster-usage traces: formatþ schema. Google Inc., White Paper; 2011.
• [3] Legrain A, Fortin M-A, Lahrichi N, Rousseau L-M. Online stochastic optimization of radiotherapy patient scheduling. In: Health care management science; 2014. p. 1–14
References
• [4] Coy P. The secret to google's success. ⟨http://www.bloomberg.com/bw/stories/ 2006-03-05/
the-secret-to-googles-success⟩; 5 March 2006 [accessed16.02.15].
• [5] Google, 2014. 2014 financial tables.
⟨https://investor.google.com/financial/tables.html⟩ [accessed 16.02.2015].
• [6] Antoine Legrain , Patrick Jaillet A stochastic
algorithm for online bipartite resource allocation problems.
Computers & Operations Research 75 (2016) 28–37,
Thank you