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.

MULTIPLE INTENTS RE-RANKING

By:

Yossi Azar, Iftah Gamzu, Xiaoxin Yin

pp. 669-678, in Proceedings of STOC 2009

Presented By:

Bhawana Goel

WEB SEARCH AND RANKING

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”

INTENT

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

INTR

OD

UC

TIO

N

PROBLEM STATEMENT

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

OPTIMAL ORDERING?

1 2 3 321

1 1 2 32 3Minimize average effort for all User types

TYPES OF INTENTS

Navigational First result is relevant

Informational All the results are relevant

Complex First and third results are relevant

OVERVIEW

Each user type has its own profile vector with subset of relevant pages <1,0…0> , <0,0…1> , <1,1…1> 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

<1,0,0> <100,0,0>One user 100 users

CALCULATION OF USER EFFORT

Navigational (<1,0,0>)2 * 1 = 2

Informational (<1,1,1>)2*1 + 4*1 + 5*1 = 11

Complex (<0.4,0.4,0.2>)2*0.4 + 4*0.4 + 5*0.2 = 3.4

1 2 3

2

4

1

9

3

1

2

3

5

4

Profile Vectors

PROBLEM FORMULATION

Form a weighted hypergraph With vertices = web results Hyperedges = user types Weights = user profiles

1 2 3

2

4

1

9

3

1

2

3

5

4

9

4

e2(1,2,3)*<1,0,0> = 1

e1(2,4,5)*<15,20,25> = 235

e2

e1

Overhead

SPECIAL CASES All user profiles are of type <1,0,…0>

It’s a case of min-sum set cover problem Its NP-hard Has an approximation ratio of 4

A B C F G IC A B

A F C B G I

Greedily pick the element which covers the most number of uncovered sets.

SPECIAL CASES All user profiles are of type <0,0,…1>

It’s a case of minimum-latency set cover problem Its NP-hard Has e-approximation algorithm

CASE 1: NON-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 F G

(4,1,0)

(3,0)

(2,2,0)

A

A F

GREEDY ALGORITHM IN GENERAL CASE

Greedy weight reduction algorithm does not work in the general case

Approximation ratio is unbounded

OPT = k2

2w + (3+4…k+2)

ALG = k3

(1+2…k) + (k+2)w

k x <1,0>

w = k2

<0,w>

CASE 2: ARBITRARY WEIGHT VECTORSHARMONIC 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 <1,0> <w/2,w>

HARMONIC INTERPOLATION

1, , ) (( )1

) (r

jr i

j i

ww w e

jw e w

i

Algorithm Phase I:1. Calculate harmonic interpolation for weight vectors for all e

e E

Algorithm Phase II:2. Calculate the weight of each vertex according to changed weight vectors3. Select vertex with maximum weight

(GREEDY WEIGHT REDUCTION ALGORITHM)

ANALYSIS OF HARMONIC INTERPOLATION ALGORITHM

Use indicator vectors :<0,0,…w…0,0> Only one entry is non-zero

Harmonic interpolation : <w/j,…w/2,w,…0> 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

OPTIMAL 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

HISTOGRAM 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

APPROXIMATION RATIO

o Reduce width of ALG by 2Hr and height by 2o The new histogram completely fits inside

optimal solution histogramo ALG/4Hr >= OPT

ALG/4

CONCLUSION

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