1. M ULTIPLE I NTENTS R E - RANKING By: Yossi Azar, Iftah Gamzu, Xiaoxin Yin pp. 669-678, in Proceedings of STOC 2009 Presented By: Bhawana Goel

2. W EB SEARCH AND R ANKING Ranking of search results on the basis of: Hyperlink structure of the web Content of the web page User’s location Not much research on user’s “intent”

3. I NTENT Same query different intents “computer science at A&M” Information about computer science department at A&M Information about admission to computer science department at A&M

4. I NTRODUCTION

5. P ROBLEM S TATEMENT 20% of web queries are ambiguous Different user types with different intents Goal is to minimize the average effort of browsing through the search results Re-rank the web results

6. O PTIMAL ORDERING ? 123 3 21 1 1 2 3 2 3 Minimize average effort for all User types

7. T YPES OF I NTENTS Navigational First result is relevant Informational All the results are relevant Complex First and third results are relevant

8. O VERVIEW Each user type has its own profile vector with subset of relevant pages,, The elements in vector correspond to positions and not particular page Order of result pages in vector is irrelevant and is determined by search engine Depicts intention Type of query need Depicts proportion of users One user100 users

9. C ALCULATION OF USER EFFORT Navigational ( ) 2 * 1 = 2 Informational ( ) 2*1 + 4*1 + 5*1 = 11 Complex ( ) 2*0.4 + 4*0.4 + 5*0.2 = 3.4 12 3 2 4 1 9 3 1 2 3 5 4 Profile Vectors

10. P ROBLEM FORMULATION Form a weighted hypergraph With vertices = web results Hyperedges = user types Weights = user profiles 1 23 2 4 1 9 3 1 2 3 5 4 9 4 e2 (1,2,3)* = 1 e1 (2,4,5)* = 235 e2 e1 Overhead

11. S PECIAL C ASES All user profiles are of type It’s a case of min-sum set cover problem Its NP-hard Has an approximation ratio of 4 A B C F G I C A B A F C B G I Greedily pick the element which covers the most number of uncovered sets.

12. S PECIAL C ASES All user profiles are of type It’s a case of minimum-latency set cover problem Its NP-hard Has e-approximation algorithm

13. C ASE 1: N ON - INCREASING WEIGHT VECTORS Non-increasing weight vectors Generalization for min-sum set cover problem Greedy weight reduction algorithm Approximation ratio of 4 A B C D E D E F G (4,1,0) (3,0) (2,2,0) A A F

14. G REEDY ALGORITHM IN G ENERAL C ASE Greedy weight reduction algorithm does not work in the general case Approximation ratio is unbounded OPT = k 2 2w + (3+4…k+2) ALG = k 3 (1+2…k) + (k+2)w k x w = k 2

15. C ASE 2: A RBITRARY WEIGHT VECTORS H ARMONIC INTERPOLATION ALGORITHM Greedy algorithm takes only local maxima into account Apply greedy algorithm on harmonically interpolated weight vectors It provides knowledge about future weight reduction potentials of hyperedges ALG = 2w/2 + (3+4…k+2) k x

16. H ARMONIC I NTERPOLATION Algorithm Phase I: 1.Calculate harmonic interpolation for weight vectors for all e  E Algorithm Phase II: 2. Calculate the weight of each vertex according to changed weight vectors 3. Select vertex with maximum weight (GREEDY WEIGHT REDUCTION ALGORITHM)

17. A NALYSIS OF HARMONIC INTERPOLATION ALGORITHM Use indicator vectors : Only one entry is non-zero Harmonic interpolation : Notations (e,i): a potential pair w(e,i): weight of the potential pair let t be the time when (e,i) is covered Penalty of a step = remaining harmonic weight/weight covered have to minimize: ∑ t=1 ∑ (e,i) w(e,i) × t

18. O PTIMAL SOLUTION HISTOGRAM Create a histogram with no of columns = number of potential pairs, width of a column = w(e,i) and height of the column = t(e,i) potential pairs  Its monotonically increasing Time

19. H ISTOGRAM FOR ALGORITHMIC SOLUTION Its not monotonic Histogram with no of columns = number of potential pairs, width of a column = ŵ(e,i) and height of the column = penalty of the step

20. A PPROXIMATION R ATIO o Reduce width of ALG by 2Hr and height by 2 o The new histogram completely fits inside optimal solution histogram o ALG/4H r >= OPT ALG/4

21. C ONCLUSION O(log r) solution is general case using harmonic interpolation and greedy algorithms Intents for all user types taken care of Better solution exists : In general case, randomized 485-approximation algorithm by Nikhil Bansal et. al. Based on stricter LP relaxation Randomized rounding