Maximizing the Relevant Diversity of Social Swarming Information

Peter Terlecky

, Yurong Jiang

, Xing Xu

, Amotz Bar-Noy

and Ramesh Govindan

CUNY Graduate Center, New York, NY, USA

University of Southern California, Los Angeles, CA, U.S.A.

Keywords:

Information Diversity, Social Swarming, Maximum Coverage.

Abstract:

In social swarming applications, users are equipped with smartphones and generate data on speciﬁc tasks

in the form pictures, video, audio, text. A central commander would like to gain access to data relevant

to a particular query. Which data wirelessly uploaded to the commander maximizes the amount of diverse

information received subject to a bandwidth constraint? We model such a problem in two distinct ways. It

is ﬁrst modeled as a maximum coverage with group budget constraints problem and then as a variant of the

maximum edge-weighted clique problem. It is shown that the algorithm for the maximum coverage model

outperforms a heuristic for the clique-based model theoretically and practically, with both performing very

well experimentally compared to an upper bound benchmark.

1 INTRODUCTION

In social swarming applications, users are equipped

with smartphones with which they generate data on

speciﬁc tasks. The data is in the form of text, video,

audio, and pictures. A central commander would like

to wirelessly receive information relevant to a query

from each of the users that is collectively comprehen-

sive.

Furthermore, he does not wish the users ﬂood the

network with all of their data, so he requires each

user to send a limited amount of the relevant collected

data. The commander would like to avoid duplicate

data or data that is similar in nature. That is, the com-

mander would like to maximize the dissimilarity of

the received data set, but have the data set be relevant

to the query/task. Which data should be uploaded

from which users to maximize the diversity of infor-

mation while maintaining bandwidth constraints?

One way in which we model such a problem is

as a maximum coverage with group budgets prob-

lem. Chekuri and Kumar (Chekuri and Kumar, 2004)

introduce and analyze the maximum coverage with

group budgets problem. They give a 2-approximation

for the cardinality version of the problem and a 12-

approximation for the cost version.

We also be model such a problem as a variant of

the maximum edge-weighted clique problem. The

maximum edge-weighted clique problem was stud-

ied in(Dijkhuizen and Faigle, 1993; Macambira and

De Souza, 2000; Hunting et al., 2001; Park et al.,

1996).

Receiving a diverse set of high quality informa-

tion is important in many applications. Consider a

shopper is deciding on whether to buy a product. The

shopper would like to receive reviews on a product.

The reviews may be grouped by category or rating. In

particular, the shopper would like to receive a diverse

subset of high quality reviews avoiding duplicate or

similar reviews. Which subset of reviews should be

shown to the shopper? Tsaparas et al. (Tsaparas et al.,

2011) focus on the problem of selecting a comprehen-

sive set of high-quality reviews covering many differ-

ent aspects of the reviewed problem. They model the

problem as a maximum coverage problem and pro-

vide algorithms which they test on a user study of

the Amazon Mechanical Turk. Lappas et al. (Lap-

pas et al., 2012) seek to select a subset of reviews

that maintain the statistical properties of the review

corpus. Yu et al. (Yu et al., 2013) seek to select a

small subset of high quality reviews which are opin-

ion diversiﬁed and cover a large set of attributes.

Zhuang et al (Zhuang et al., 2006) consider the prob-

lem of extracting the features on which movie review-

ers express their opinions and determining whether

the opinions are positive or negative. They propose

an effective multi-knowledge based solution.

Chen et al. (Chen et al., 1997) consider the prob-

lem of ﬁnding an optimal subset of features, that is,

from a large set of candidate features, selecting a sub-

365

Terlecky P., Jiang Y., Xu X., Bar-Noy A. and Govindan R..

Maximizing the Relevant Diversity of Social Swarming Information.

DOI: 10.5220/0004717503650372

In Proceedings of the 3rd International Conference on Sensor Networks (SENSORNETS-2014), pages 365-372

ISBN: 978-989-758-001-7

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

set of features which are able to represent given ex-

amples (samples) consistently. They proves that the

problem of ﬁnding an optimal subset of features is

NP-hard, and presents a heuristic for solution.

Margules et al. (Margules et al., 1988) considers

the problem of selecting a set of vendors in the man-

ufacturing environment. This paper proposes a deci-

sion support approach to selecting vendors under the

conﬂicting criteria of minimizing the annual material

costs, reducing the number of suppliers and maximiz-

ing suppliers’ delivery and quality performances.

The context of our work is social swarming. This

context has also been the setting for the following

works. Liu et al. (Liu et al., 2012a) considered max-

imizing the credibility of social swarming informa-

tion. In (Liu et al., 2012b), Liu et al. were intersted

in maximizing the number of timely reports sent to

a commander. Jiang et al. (Jiang et al., 2013) devel-

oped a system called Mediascope for selective timely

retrieval of media from mobile devices.

2 MODEL

A commander has m reporters in the ﬁeld collecting

data in various ways: photos, video, sound record-

ings, text. The commander would like to receive as

much information as possible on a particular event or

cirumstance. He would like the reporters to upload

some subset of their reports so that the total “infor-

mation” collected is maximized. Each report has a

particular size and there is a limited amount of band-

width assigned to each reporter. Given bandwidth

constraints, he would like to determine which reports

should be uploaded so as to maximize the total infor-

mation received. Here information is modeled as tags

which the reporters set to the reports. For example,

if a reporter takes a photo, he tags the images in the

photo. If a reporter reports via text, he includes key-

words as his tags. The tags are the elements in the

information universe I.

2.1 Single Format

First, let us assume all reports are of one format, and

let us assume that this format is photos. The infor-

mation universe is comprised of the elements in the

set I = {i

,...,i

}. A photo or information set J is

a subset of I. Assume there are m users. User k’s

smartphone has u

photos. Denote the photo set by

. Let photo p

k j

denote photo j from user k. We wish

to select s

≤ u

photo’s from user k, k ∈ {1,...,m}

such that we maximize the information obtained, i.e.

the cardinality of the union of the information sets se-

lected. We may assume without loss of generality that

= 1 for each k for if s

> 1: we may make s

copies

of user k’s photo set and select one photo from each

copy and if s

= 0, we may ignore the photo set. We

call this problem the Single Format Maximum Infor-

mation Coverage Problem (SFINFOCOVER) and can

represent this problem by the following program.

max | ∪ p

k j

s.t.

∑

j=1

k j

≤ 1 k ∈ {1,. . . , m} (1)

k j

∈ {0,1} ∀k, j

(2)

It is represented as an integer program in the follow-

ing way:

max

∑

i=1

s.t.

∑

j=1

k j

≤ 1 k ∈ {1,. . . , m} (3)

≤

∑

i∈p

k j

i ∈ {1,...,n} (4)

k j

∈ {0,1} ∀i, j,k

(5)

The variables y

, i = 1, . . . , n are indicator variables for

selecting the i-th element. The variables x

k j

are indi-

cator variables for selecting the j-th set from photo

set k. Inequality 3 assures that at most one photo is

selected from each photo set. Inequality 4 guarantees

that if no set is selected containing element i, then el-

ement i is not selected.

2.2 Multi-format

We now turn our attention to the multi-format sce-

nario. In this scenario each reporter has a set of re-

ports of different formats i.e. video, audio, text, etc.

that can be uploaded to the commander. Let R

i j

de-

note report j from reporter i and let r

= {R

,...,R

}

be the collection of u

reports from reporter i. Let I =

,...,i

} denote the information universe. Each re-

port R

i j

is a subset of the information universe R

i j

⊆ I,

for all i, j.

There is a size for each report which represents

the ﬁle size of that report. Let s(R

i j

) denote the size

of report R

i j

. Certainly reports of different formats

have different sizes. Video ﬁles tend to be much larger

than text ﬁles or photos, but they can also offer more

information. Even within a format, ﬁles can be of

different sizes. Consider video, audio, or text ﬁles.

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366

They may be of different duration or length and as the

duration or length of a ﬁle increase as does its size.

Reporter i has a ﬁxed amount of bandwidth with

which to upload reports and this is a constraint on

the size of the reports that can be uploaded by a re-

porter. Let b

denote the total size which can be

uploaded by reporter i. Given the information uni-

verse I = {i

,...,i

}, bandwidth constraints b

...,b

for reporters r

,...,r

respectively, reports {R

i j

}, and

size of reports {s(R

i j

)}, which reports should be up-

loaded to maximize the number of obtained infor-

mation elements subject to the bandwidth constraints

...,b

This problem is called the Multi-Format

Maximum Information Coverage Problem

(MFINFOCOVER) and it can be represented by

the following IP.

max

∑

i=1

s.t.

∑

t=1

s(R

≤ b

i ∈ {1,...,m} (6)

≤

∑

i∈R

k j

i ∈ {1,...,n}

k j

∈ {0,1} ∀i, j,k

(7)

The signiﬁcant difference of the IP for MFINFO-

COVER compared to the IP for SFINFOCOVER is in-

equality 6 which bounds the sum of the sizes of re-

ports uploaded by a reporter by the bandwidth avail-

able for that reporter.

2.3 Clique Model

In this subsection, we present the clique model. As-

sume the information universe is comprised of the el-

ements in the set I = {i

,...,i

}. Let R

i j

denote re-

port j from reporter i. In particular, R

i j

⊆ I. Rep-

resent each report R

i j

∀i, j by a vertex. Each ver-

tex R

i j

has a size denoted by s(R

i j

). The set of ver-

tices is partitioned into m classes r

,...,r

, where a

class represents a reporter’s set of reports. That is,

= {R

,...,R

} where u

is the number of reports

of reporter i. Each class r

has a capacity b

, cor-

responding to the bandwidth allocated to a reporter

to upload his reports. This report graph is a com-

plete graph with edge e

it, jk

having weight w

it, jk

. The

weight of an edge is the symmetric difference of the

information sets of the vertices connected by the edge.

That is,

it, jk

= |R

∪ R

| − |R

∩ R

| (8)

This weight measures how distinct the two reports

are by counting the number of elements they differ

in collectively. The optimization objective is to select

a sub-clique of vertices in which the sum of the edge-

weights is maximum over all feasible sub-cliques. A

sub-clique is feasible if the sum of the sizes of all

vertices selected from a class is at most the capacity

of the class, for every class, i.e.

∑

t=1

s(R

≤ b

∀i = 1,...,m. We call this problem the Clique Infor-

mation Coverage Problem (CLIQUEINFOCOVER).

We can formulate CLIQUEINFOCOVER as the fol-

lowing integer program.

max

∑

it6= jk

it, jk

s.t. y

it, jk

≤ x

∀it 6= jk (9)

it, jk

≤ x

∀it 6= jk (10)

+ x

− y

it, jk

≤ 1 ∀it 6= jk (11)

∑

t=1

s(R

≤ b

∀i = 1, . ..,m (12)

∈ {0,1} ∀i

it, jk

∈ {0,1} ∀it 6= jk

In the integer program, variable x

is used as

the indicator variable for selecting vertex R

for the

clique. Similarly, variable y

it, jk

is used as the indi-

cator variable for selecting edge e

it, jk

for the clique.

Inequalities 9 and 10 ensure that edge e

it, jk

is not se-

lected if either vertex it or jk is not selected. Inequal-

ity 11 guarantees that y

it, jk

is selected if both both ver-

tices it and jk are selected. Inequality 12 ensures that

the total size of vertices chosen from class r

does not

exceed b

We analyze the integrality gap of the above integer

program. Consider the following example with n an

even integer. There are two classes, with b

= b

= 1

and I = {1, . . . , n}. Class 1 consists of the reports

{{1},{2},...,{n/2}} with each report being size 1.

Class 2 consists of the reports {{n/2 + 1}, . . . , {n}}

with each report also being of size 1. Note that any

two reports are disjoint. The optimal integer program-

ming solution chooses any set from class 1 and any

set from class 2 and obtains an objective value of 2. A

linear programming solution that selects x

= 1/(n/2)

for all i obtains an objective of 2 ·





· 1/(n/2) which

is 2(n − 1). Thus, the integrality gap of such an inte-

ger program is at least n − 1. This result implies that

rounding a linear programming relaxation is not very

amenable for solution and that a linear programming

upper bound on the optimal could be a very loose

upper bound. With this in mind, we also consider

a quadratic programming formulation of CLIQUEIN-

FOCOVER.

MaximizingtheRelevantDiversityofSocialSwarmingInformation

367

The following is a quadratic programming for-

mulation of CLIQUEINFOCOVER, it is similar to the

quadratic program presented in (Alidaee et al., 2007)

max

∑

it6= jk

it, jk

s.t.

∑

t=1

s(R

≤ b

∀i = 1, . ..,m

∈ {0,1} ∀it

The intuition behind the optimization goal of se-

lecting a sub-clique with largest sum of edge-weights

is to select a subset of reports which have high pair-

wise distinction with the belief that this will maxi-

mize the total number of elements covered. Consider

the following instance: Let I = {1,...,n}, reporter 1

has reports {1,...,n} and {bn/2c − 1} each of size

1 and reporter 2 has report {bn/2c, . . . , n} of size 1.

We have that b

= b

= 1 and thus each reporter can

upload exactly one report. The reports which maxi-

mize the sum of dissimilarity are reports {bn/2c− 1}

and {bn/2c,...,n} giving a dissimilarity of bn/2c+1,

while the reports which maximize the amount of in-

formation obtained are {1,...,n} and {bn/2c, . . . , n}

as they give the full information set I. The difference

in obtained information is bn/2c − 2 elements.

3 ALGORITHMS AND

HEURISTICS

In this section we provide approximation algorithms

and heuristics for the problems. It follows that de-

ﬁned problems are NP-Hard. The hardness of SFIN-

FOCOVER and MFINFOCOVER follow from a trivial

reduction from the Maximum Coverage problem. It

also holds that CLIQUEINFOCOVER is NP-hard even

for one class with 0/1 edge weights. The hardness of

CLIQUEINFOCOVER follows from a reduction from

the Maximum Clique problem. It also holds that there

is no constant factor approximation for CLIQUEINFO-

COVER, from this reduction.

The following greedy approximation algorithm

called GREEDY for SFINFOCOVER gives a 2-

approximation. The analysis is given in Chekuri and

Kumar (Chekuri and Kumar, 2004). The idea behind

GREEDY is as follows: for each photo set from which

a photo has not already been selected, select the photo

which covers the maximum number of uncovered el-

ements, breaking ties arbitrarily. For a given round,

select the photo out of these maximal photos which

covers the maximum number of uncovered elements.

Remove these elements from consideration in future

rounds, and remove this photo set from consideration

in future rounds.

Algorithm 1: GREEDY.

1: H ←

0, I

← I, S ←

2: while S 6= {1,.. .,m} and I

0 do

3: for all Reporters k ∈ {1,.. . ,m} do

4: if a k 6∈ S then

5: l ← argmax

{|p

k j

∩ I

6: G

← p

7: else

8: G

←

9: end if

10: end for

11: c ← argmax

12: H ← H ∪ {G

}, S ← S ∪ {c}, I

← I

− G

13: end while

OUTPUT: H, I − I

Next, we present an approximation algorithm for

MFINFOCOVER called MFGREEDY. It is a 12-

approximate algorithm (Chekuri and Kumar, 2004).

This algorithm adds reports greedily, in the following

sense. In a given round, it computes the coverage per

size of every remaining feasible report. It adds a fea-

sible report with the largest coverage per size.

Algorithm 2: MFGREEDY.

1: H ←

0, I

← I, b

← b

∀k ∈ {1, ..., m}, G

= R

2: while G

0 do

3: for all Reporters k ∈ {1,.. . ,m} do

4: for all reports R

5: if s(R

) > b

then

6: R

←

7: end if

8: l ← argmax

{|R

k j

∩ I

|/s(R

k j

)}

9: G

← R

10: end for

11: end for

12: c ← argmax

∩ I

|/s(G

)

13: H ← H ∪ {G

}, I

← I

− G

, b

← b

− s(G

)

14: end while

OUTPUT: H, I − I

We propose the following heuristic for CLIQUE-

INFOCOVER which we call CLIQUE-MAXSUM. For

each reporter i, for each report R

, t ∈ {1,. . . , u

} be-

longing to reporter i, compute the ratio of sum of edge

weights to size of the report. Until capacity b

would

be exceeded, add the feasible report with largest ratio

from reporter i (and remove this report from r

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368

Algorithm 3: CLIQUE-MAXSUM.

1: C ←

2: for all reporters i ∈ {1,.. .,m} do

3: for all Reports R

, t ∈ {1, ... ,u

} do

4: S

←

∑

it, jk

5: Compute the ratio k

= S

/s(R

)

6: end for

7: while there is a report which can be added within

capacity b

8: C ← C ∪ {feasible report R

∗

with maximum k

}

9: Remove R

∗

from r

10: end while

11: end for

OUTPUT: C and sum of edge-weights of C

4 SIMULATIONS

In the simulation environment, we are interested in

comparing the performance of the SFINFOCOVER

model to the CLIQUEINFOCOVER model. We evalu-

ate the CLIQUE-MAXSUM (abbreviated by CLIQUE)

and GREEDY algorithms as well as an LP-relaxation

(LP) for the SFINFOCOVER IP for randomly gener-

ated instances. LP acts as an upper bound on the op-

timal solution and is therefore a good scalable bench-

mark for comparison. The following parameters are

varied in the simulations: the number of reporters,

number of elements n in the information universe I,

the number of photos per reporter, and the probability

p that an element is in a photo.

In Simulation 1, we set the number of reporters to

3 with each reporter have 3 photos. The number of

items is varied from 1000 to 1050 and for each n 10

runs are performed. In a run, for each photo, an item

appears with probability 0.5. GREEDY outperforms

CLIQUE on average by about 5 items.

In Simulation 2, the number of reporters is 4, and

each reporter has 2 photos from which at most one can

be selected. The number of items varies from 2000 to

2100 with 10 random runs being performed for each

n. In a run, for each photo, an item appears with

probability 0.5. GREEDY outperforms CLIQUE with

GREEDY covering on average about 96% of items.

In Simulation 3 the number of reporters is varied

from 1 to 5. There are 20 items and 2 photos per

reporter. With both the GREEDY and CLIQUE algo-

rithms, the number of items obtained increases as the

number of reporters increases. GREEDY outperforms

CLIQUE in the number of items selected with both

algorithms converging to covering all 20 items when

n = 5. For a run, an item of a photo has a probability

of .5 of appearing.

In Simulation 4, the total number of items is 40,

and there are two photos per reporter. The number

of reporters varies from 1 to 6. GREEDY outperforms

CLIQUE in the number of items selected, but the out-

performance is minor. We see that when there is 1

reporter both algorithms cover on average 20 items.

This is expected as an item for a photo has a probabil-

ity of .5 of appearing.

In Simulation 5, the number of items is varied

from 20 to 40. There are 3 reporters and 2 photos per

reporter. The linear programming relaxation of the IP

is also simulated along with GREEDY and CLIQUE.

GREEDY and CLIQUE perform quite well to the LP

which is an upper bound on the actual IP solution.

GREEDY yet again outperforms CLIQUE, but the out-

performance is quite minor. There were 15 runs for

each n, and for a run, an item of a photo has a proba-

bility of .5 of appearing.

In Simulation 6, we set the number of reporters to

3 with 2 photos per reporter. The number of items

is varied from 100 to 140. A photo matrix is ran-

domly generate with the probability of an item ap-

pearing in a photo of .5. All three algorithms are run

twenty times for a given n. GREEDY once again out-

performs CLIQUE, with GREEDY being on average

about 4 items from the LP upperbound on the opti-

mal.

In Simulation 7, we vary the number of photos per

reporter from 1 to 10. We hold constant the number of

reporters at 3 and the number of items at 20. The num-

ber of runs for each p for each algorithm is 30. We see

that the performance of CLIQUE falls off drastically

from the performance of the LP and GREEDY. The

performance of CLIQUE maintains a mean of about

15 over all p, where both LP and GREEDY increase as

p increases with GREEDY covering on average about

18 items and LP covering about 20 with p = 10.

In Simulation 8 we vary n from 100 to 140 setting

the probability of an item being in a photo to be .2.

For each n, there is 100 runs for each of the 3 algo-

rithms. The number of reporters is 3 and the number

of photos per reporter is 2. GREEDY and CLIQUE

perform comparably with CLIQUE for some n barely

outperforming GREEDY on average. Both algorithms

select on average about 2 items less than the LP.

In Simulation 9, we vary the probability of an item

being included in a photo from 0 to 1 by 0.05. We

hold ﬁxed the number of items at 4, the number of

reporters at 4, and the number of photos per reporter

at 3. We see that both LP and GREEDY converge to

covering all 4 items at a probability of .4. CLIQUE

takes a longer time to converge to 4. It converges at a

probability of .8.

In Simulation 10, the probability that an item is

included in a photo is varied from 0 to 1 by 0.05.

The number of items is 10, the number of reporters

MaximizingtheRelevantDiversityofSocialSwarmingInformation

369

1000 1005 1010 1015 1020 1025 1030 1035 1040 1045 1050

970

980

990

1000

1010

1020

1030

# of elements covered

Simulation with m=5, p=3, #runs for each n is 10, photo signiture uniformly generated

greedy

clique

Figure 1: Simulation 1.

2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

1860

1880

1900

1920

1940

1960

1980

2000

# of elements covered

Simulation with m=4, p=2, number of runs=10, photo signature uniformly generated

greedy

clique

Figure 2: Simulation 2.

1 1.5 2 2.5 3 3.5 4 4.5 5

# of elements covered

Simulation with n=20, p=2, photo signature uniformly generated

greedy

clique

Figure 3: Simulation 3.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

# of elements covered

Simulation with n=40, p=2, photo signature uniformly generated

greedy

clique

Figure 4: Simulation 4.

20 22 24 26 28 30 32 34 36 38 40

# of elements covered

Simulation with m=3, p=2

greedy

clique

lp−set

Figure 5: Simulation 5.

100 105 110 115 120 125 130 135 140

100

105

110

115

120

125

130

135

# of elements covered

Simulation with m=3, p=2

greedy

clique

lp−set

Figure 6: Simulation 6.

1 2 3 4 5 6 7 8 9 10

15.5

16.5

17.5

18.5

19.5

# of elements covered

Simulation where n=20, m=2, #runs=30 for each p

greedy

clique

lp−set

Figure 7: Simulation 7.

100 105 110 115 120 125 130 135 140

# of elements covered

100 runs, pr=.2

greedy

clique

lp−set

Figure 8: Simulation 8.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5

1.5

2.5

3.5

# of elements covered

n=4,p=3,m=4, Probability that an item is in a set varies from 0 to 1

greedy

clique

lp−set

Figure 9: Simulation 9.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

# of elements covered

n=10,p=4,m=4,nruns=15 for each probability

greedy

clique

lp−set

Figure 10: Simulation 10.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

# of elements covered

n=3−,p=3,m=4, #runs for each pr=15

greedy

clique

lp−set

Figure 11: Simulation 11.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

# of elements covered

m=3,p=3, n=30, number of runs=15

greedy

clique

lp−set

Figure 12: Simulation 12.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

# of elements covered

n=80, m=8,p=10

greedy

clique

lp−set

Figure 13: Simulation 13.

1 2 3 4 5 6 7 8 9 10

# of elements covered

n=30, m=2, #runs=30

greedy

clique

lp−set

Figure 14: Simulation 14.

is 4, and there are 4 photos per reporter. The num-

ber of runs for each probability is 15. We see that the

performance of GREEDY is quite close to the perfor-

mance of LP. LP converges to covering all items at a

probability of about 0.4. GREEDY converges to cov-

ering all items at a probability of about .5. CLIQUE

converges to covering all items at a probability of .9.

CLIQUEs performance matches LP and GREEDY for

probabilities of 0 to .3, but then its performance falls

off slightly.

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In Simulation 11, we set the number of items to be

covered to 30. The number of reporters is 4 and each

reporter has 3 photos from which he needs to choose

one from. The probability of a item being included

in a photo varies from 0 to 1 by .05. The number of

runs for each probability is 15. LP converges to cov-

ering all items at a probabiliy of about .5. GREEDY

converges to covering all items at a probability of .6.

CLIQUE converges to covering all items at a probabil-

ity of 0.85. While its performance is on par with LP

and GREEDY when the probability is between 0 and

.4, it underperforms both from .4 to .85.

In Simulation 12, we set the number of items to be

covered to 30. The number of reporters is 3 and each

reporter has 3 photos from which he needs to choose

one from. The probability of a item being included

in a photo varies from 0 to 1 by .05. The number of

runs for each probability is 15. LP converges to cov-

ering all items at a probabiliy of about .5. GREEDY

converges to covering all items at a probability of .6.

CLIQUE converges to covering all items at a probabil-

ity of 0.85. While its performance is on par with LP

and GREEDY when the probability is between 0 and

.4, it underperforms both from .4 to .85.

In Simulation 13, we once again vary probability

that an item is included in a photo from 0 to 1 by

.05. This time there are 80 total items, the number of

reporters is 8 and the number of photos per reporter

is 10. All three algorithms converge to covering all

items rather quickly, with LP converging at a proba-

bility of about .15, GREEDY converging at a probabil-

ity of about .25 and CLIQUE converging at a probabil-

ity of about .4. A large gap of about 7 items on aver-

age can be seen in this simulation between GREEDY

and CLIQUE at a probability of about 0.2.

In Simulation 14, we vary the number of photos

per reporter. We ﬁx the number of reporters to 2 and

the number of items to be covered to 30. We set the

probability of an item appearing in a photo to be .6.

We run each of the three algorithms 30 times for a

given number of photos per reporter. We see the per-

formance of CLIQUE falls off signiﬁcantly in com-

parison to the other two algorithms as p increases, se-

lecting on average 23 items over all p, and remaining

relatively ﬂat on average.

From these simulations, we see that GREEDY

outperforms CLIQUE with GREEDY performing very

well compared to an LP-relaxation upper bound on

the optimal solution.

5 CONCLUSIONS AND FUTURE

WORK

We propose two novel models, a maximum cover-

age based model and a clique based model for mod-

eling the problem of maximizing the amount of rel-

evant diversity of social swarming data received by

a commander. We provide well-performing algo-

rithms/heuristics for both models with the maximum

coverage algorithm outperforming the clique heuris-

tic. In future work, we look to experimentally ana-

lyze the multiple-format models and develop a physi-

cal system based on our models.

ACKNOWLEDGEMENTS

Research was sponsored by the Army Research Lab-

oratory and was accomplished under Cooperative

Agreement Number W911NF-09-2-0053. The views

and conclusions contained in this document are those

of the authors and should not be interpreted as rep-

resenting the ofﬁcial policies, either expressed or im-

plied, of the Army Research Laboratory or the U.S.

Government. The U.S. Government is authorized

to reproduce and distribute reprints for Government

purposes notwithstanding any copyright notation here

on.

REFERENCES

Alidaee, B., Glover, F., Kochenberger, G., and Wang,

H. (2007). Solving the maximum edge weight

clique problem via unconstrained quadratic program-

ming. European journal of operational research,

181(2):592–597.

Chekuri, C. and Kumar, A. (2004). Maximum cover-

age problem with group budget constraints and ap-

plications. Approximation, Randomization, and Com-

binatorial Optimization. Algorithms and Techniques,

pages 72–83.

Chen, B., HONG, J., and WANG, Y. (1997). The prob-

lem of ﬁnding optimal subset of features. CHINESE

JOURNAL OF COMPUTERS-CHINESE EDITION-,

20:133–138.

Dijkhuizen, G. and Faigle, U. (1993). A cutting-plane ap-

proach to the edge-weighted maximal clique prob-

lem. European Journal of Operational Research,

69(1):121–130.

Hunting, M., Faigle, U., and Kern, W. (2001). A lagrangian

relaxation approach to the edge-weighted clique prob-

lem. European Journal of Operational Research,

131(1):119–131.

Jiang, Y., Xu, X., Terlecky, P., Abdelzaher, T. F., Bar-Noy,

A., and Govindan, R. (2013). Mediascope: selective

MaximizingtheRelevantDiversityofSocialSwarmingInformation

371

on-demand media retrieval from mobile devices. In

IPSN, pages 289–300.

Lappas, T., Crovella, M., and Terzi, E. (2012). Select-

ing a characteristic set of reviews. In Proceedings

of the 18th ACM SIGKDD international conference

on Knowledge discovery and data mining, pages 832–

840. ACM.

Liu, B., Terlecky, P., Bar-Noy, A., Govindan, R., Neely,

M. J., and Rawitz, D. (2012a). Optimizing infor-

mation credibility in social swarming applications.

IEEE Transactions on Parallel and Distributed Sys-

tems, 23(6):1147–1158.

Liu, B., Terlecky, P., Xu, X., Bar-Noy, A., Govindan, R.,

and Rawitz, D. (2012b). Timely report delivery in so-

cial swarming applications. In DCOSS, pages 75–82.

Macambira, E. and De Souza, C. (2000). The edge-

weighted clique problem: valid inequalities, facets

and polyhedral computations. European Journal of

Operational Research, 123(2):346–371.

Margules, C., Nicholls, A., and Pressey, R. (1988). Select-

ing networks of reserves to maximise biological diver-

sity. Biological conservation, 43(1):63–76.

Park, K., Lee, K., and Park, S. (1996). An extended formu-

lation approach to the edge-weighted maximal clique

problem. European Journal of Operational Research,

95(3):671–682.

Tsaparas, P., Ntoulas, A., and Terzi, E. (2011). Select-

ing a comprehensive set of reviews. In Proceedings

of the 17th ACM SIGKDD international conference

on Knowledge discovery and data mining, pages 168–

176. ACM.

Yu, W., Zhang, R., He, X., and Sha, C. (2013). Selecting

a diversiﬁed set of reviews. In Web Technologies and

Applications, volume 7808 of Lecture Notes in Com-

puter Science, pages 721–733.

Zhuang, L., Jing, F., and Zhu, X.-Y. (2006). Movie re-

view mining and summarization. In Proceedings of

the 15th ACM international conference on Informa-

tion and knowledge management, CIKM ’06, pages

43–50.

SENSORNETS2014-InternationalConferenceonSensorNetworks

372