MULTIPLE INTENTS RE-RANKING

By:Yossi Azar, Iftah Gamzu, Xiaoxin Yinpp. 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

INTRO

DU

CTION

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 EFFORTNavigational (<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> = 235e2

e1Overhead

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 IGreedily 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 HISTOGRAMCreate 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 RATIOo 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