REGION-BASED HEURISTICS FOR AN ITERATIVE

PARTITIONING PROBLEM IN MULTIAGENT SYSTEMS

Thomas Kemmerich

International Graduate School Dynamic Intelligent Systems, University of Paderborn, 33095 Paderborn, Germany

Hans Kleine B

uning

Department of Computer Science, University of Paderborn, 33095 Paderborn, Germany

Keywords:

Region-based heuristic, Partitioning, Multi-objective optimization, Multiagent system, Coordination, External

storage media.

Abstract:

Load balancing or access point selection in wireless networks both are problems where a large set of particles

repeatedly has to be partitioned on another set of objects. In general this partitioning problem involves multiple

contrary objectives. Due to the large number of particles a decentralized approach should be favored. In this

work, such an iterative multi-objective optimization problem is modeled as multiagent system. We propose a

local solution technique based on regions and some special coordination media. Agents select target objects

based on the region they are in. Different region types are considered and a local heuristic is developed. We

show the general potential of regions and experimentally analyze different approaches. All approaches are

able to provide high quality solutions.

1 INTRODUCTION

Repeated partitioning of some elements onto another

set of special elements is a frequently occurring prob-

lem in a variety of different real systems. For instance

when multiple users have to be allocated to some

servers. Another example can be found in Wireless

LANs, where mobile devices have to select an access

point (AP) from a set of reachable APs. In the latter

system, recent AP selection protocols still are unable

to solve the problem appropriately in all settings (Yen

et al., 2009). The used protocols basically make a

mobile device to independently select the AP with the

strongest signal to noise ratio. Since bandwidth at an

AP has to be shared among all participants, one can

easily imagine settings where such simple protocols

will result in unbalanced assignments of devices to

access points, esp. if large numbers of devices are in-

volved. Several works deal with this issue in WLANs,

e.g. (Yen et al., 2009), or (Kasbekar et al., 2006).

In this work we consider such a repeated or iter-

ative partitioning problem that involves multiple op-

Note that the presented approaches may not directly be trans-

fered to real systems as load balancing has to take into account

trafﬁc patterns, too. However, our results could be considered in

the design of novel AP selection mechanisms.

timization criteria. We model it as dynamic multi-

agent system (MAS) where agents change positions

over time and agents as well as mechanisms should

be as simple as possible. Based on local information,

agents have to partition themselves onto a set of target

objects. The created partitionings are supposed to be

optimized against some contrary objectives.

Up to now, even no efﬁcient central approach to

solve this general problem in scenarios with more

than two targets and an arbitrary number of agents is

known (Goebels, 2007). We conjecture that the prob-

lem is at least NP-hard. Hence, efﬁcient heuristics

have to be developed. As we are interested in local

algorithms in that ﬁeld, we propose to approach the

problem using regions that are deﬁned based on tar-

get positions. Information about these regions will be

used locally by agents to determine a target assign-

ment. At the same time these local decisions should

lead to good partitionings according to global objec-

tives. Besides our focus on local approaches, we are

also interested in building simple algorithms which

only involve very simple knowledge structures and

reasoning processes. The question is how simple both

can become on the one hand while on the other hand

they still should enable agents to ﬁnd good solutions.

Section 2 brieﬂy deals with related work. Sec-

200

Kemmerich T. and Kleine Büning H..

REGION-BASED HEURISTICS FOR AN ITERATIVE PARTITIONING PROBLEM IN MULTIAGENT SYSTEMS.

DOI: 10.5220/0003151202000205

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 200-205

ISBN: 978-989-8425-41-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

tion 3 then presents a detailed model of the problem.

Next, we proceed stepwise to develop a local heuris-

tic to solve the given problem. We will ﬁrst deﬁne

the considered Target-Regions in Section 4. These re-

gions are deﬁned based on target positions in static

scenarios and build the basis of the heuristics devel-

oped in this paper. An experimental

investigation of

an hypothesis on the solution quality of approaches

that use Target-Regions is given. As Target-Regions

represent a good mean to solve the considered prob-

lem but at the same time are hard to calculate, we have

to approximate them for general scenarios. Therefore,

we propose an approximation technique in Section 5

that will be evaluated, too. Based on these insights

we ﬁnally develop the local approach. It will use so-

called storage media which can be considered as co-

ordination objects as they are used by agents to ex-

ternally store information about “good” target assign-

ments. The experimental results show that this local

approach is able to ﬁnd high quality solutions in gen-

eral settings. In the end, we conclude this work in

Section 6.

2 RELATED WORK

The problem underlying the iterative partitioning task

considered in this work is based on the Online Parti-

tioning Problem (Goebels, 2007). In (Ducatelle et al.,

2009), the authors solve a similar problem using a

communication-based and a reactive approach.

The iterative problem considered in this work de-

mands an assignment of agents to targets and thereby

in a sense a formation of groups. In contrast to

clustering or (iterative) graph partitioning (Fj

allstr

om,

1998), where elements are clustered according to

some metrics, an additional constraint applies in our

problem as targets have to be in different groups.

In the area of RoboCup Rescue, task allocation

algorithms are investigated, too. In (Sedaghat et al.,

2006), a simple partitioning technique that divides

a map in regular regions in combination with an

auction-based mechanism is shown to be an effective

mean to solve their task allocation problem.

The local heuristic developed in this work is

based on a framework that incorporates external stor-

age media in the reasoning process of a capacity-

constrained multiagent systems (MAS) (Kemmerich

and Kleine B

uning, 2010a). Knowledge of agents is

stored on passive external storage media (SM) that are

located within the environment. Hence, storage media

Source code, evaluation scripts, and conﬁguration ﬁles of all

experiments conducted in this work are available for download at

http://www.upb.de/cs/ag-klbue/en/staff/kemmerich/icaart10.tgz

target

communication radius of a

storage

medium

sensor/actor

radius of a

read/

write

agent

with job

(remaining

duration 5)

Figure 1: MAS with agents, jobs, targets, and external stor-

age media.

become a mean for the coordination of agents. Stor-

age media are comparable to coordination artifacts as

introduced in (Omicini et al., 2004). These artifacts

are also meant to support coordination in MAS and ﬁ-

nally resulted in the A&A meta-model (Omicini et al.,

2008). In that context, environments as ﬁrst class ab-

stractions in MAS to support coordination, coopera-

tion, and interaction between agents recently gain an

increasing interest (Parunak and Weyns, 2007).

3 PROBLEM STATEMENT

Figure 1 illustrates the considered MAS which, be-

sides using a grid environment, is basically in a line

with (Kemmerich and Kleine B

uning, 2010a). It con-

sists of a set of agents A = {a

,...,a

}, a set of tar-

gets T = {t

,...,t

}, and a set M = {m

,...,m

} of

external storage media which will be used for coor-

dination purposes later in this work. All entities are

placed in a 2-dimensional Euclidean grid environment

E as deﬁned in Deﬁnition 1.

Deﬁnition 1 (A,T -Grid-Environment)

An A,T -Grid-Environment E is a rectangular grid

environment containing |A| agents, |T | targets, and a

set of storage media M . Each object is located within

one cell of the grid, i.e. no two objects intersect. Let

c = size(E ) denote the total number of cells in the

grid. Then we call an A,T -Grid-Environment full, if

and only if c = |A| + |T |, and sparse otherwise.

Each agent works on a job located in the environ-

ment. Each job j ∈ J requires ct( j) ∈ [min

,max

]

time steps and is executed till completion. Job posi-

tions and durations are chosen uniformly at random.

If a job is done, the environment assigns the agent to

a newly created random job. Accordingly, this proce-

dure leads to dynamically changing agent positions

while target and storage media positions are ﬁxed.

Agents interact locally with storage media by reading,

writing, and deleting information items. The model

REGION-BASED HEURISTICS FOR AN ITERATIVE PARTITIONING PROBLEM IN MULTIAGENT SYSTEMS

201

further limits the internal memory capacity of agents

and storage media using a ring buffer. This implies

that information might be overridden. The neighbor-

hood N

of agent a consists of up to k agents that are

nearest to a within a given communication radius.

Let p : A → T be a total function that maps el-

ements from A to elements from T . Then a parti-

tioning of two non-empty sets A and T is deﬁned as

a multiset S

A,T

= {S

,...,S

|T |

} having S

= {a ∈

A | p(a) = t

The considered problem demands a repeated par-

titioning of agents to targets. The goal is to ﬁnd good

partitionings in each iteration using only local infor-

mation that optimize three contrary objectives:

1. create a uniform distribution, i.e. assign approx-

imately

|A|

|T |

agents to each target

2. minimize the distance sum between agents and

selected targets

3. minimize the costs that are produced according to

a cost model

To measure the quality of a partitioning estab-

lished in iteration , we use a function f : S



A,T

→

[0,1] that is calculated from a global perspective. In

this work, Equation 1 realizes this function and cal-

culates the weighted sum of the ﬁrst two normalized

objective values. Weights α and β in general must be

greater or equal to zero and sum to one. If not stated

otherwise, we will use α = β =

. Note that crite-

ria three (costs) is not considered in this formula as it

does not directly inﬂuence the pure partitioning qual-

ity. For a discussion on how costs could be modeled

and for a brief cost analysis, we refer to the extended

version of this paper (Kemmerich and Kleine B

uning,

2010b).

f (S



A,T

) = α ·







∏

∈S



A,T



|A|

|T |



|T |







+β ·







∑

a∈A

δ(a,τ(a))

∑

a∈A

δ(a, p(a))







(1)

The ﬁrst term of Equation 1 represents the distribu-

tion objective value derived from a partitioning S



A,T

It is normalized against an optimal uniform distribu-

tion. The second term normalizes the distance sum

of S



A,T

against the minimal possible distance sum.

Here, τ : A → T is used which returns the nearest

target of an agent (ties are broken by selecting the tar-

get t

with the lowest index i). The Euclidean dis-

tance between an agent and a target is calculated by

δ : A × T → R

. It is particularly noticeable that

agents cannot calculate partitioning qualities on their

own due to partial observability of the environment.

The overall goal of the agents then is to maximize

the average partitioning quality over some iterations k

as expressed by

q =

∑

=1

f (S



A,T

). (2)

4 TARGET-REGIONS

We now propose the basic mechanism upon which the

ﬁnal local heuristic will be based on. Therefore, we

ﬁrst deﬁne the term Target-Regions:

Deﬁnition 2 (Target-Region)

A Target-Region TR(t) for any target t in a full A-T -

Grid-Environment is deﬁned by a set of cells C

. The

set consists of target t’s cell and all cells of agents

that, in an optimal solution, are assigned to t.

Based on this deﬁnition, the basic idea is very simple:

given Target-Regions and a full environment, agents

create partitionings by just selecting the target of the

region they are located in. The corresponding algo-

rithm then reads as follows:

Algorithm 1: Executed by each agent a ∈ A.

1: Determine current agent position pos(a)

2: Determine Target-Region TR(t) at pos(a)

3: Assign agent a to target t

Obviously, the decision making itself cannot be

simpler besides having an build-in oracle that makes

the agents know the best target to select at any time.

Also the knowledge structure is very simple as it is

only composed of one position and one target infor-

mation item. Hence this idea ﬁts well to our intentions

of building simple algorithms.

In addition, this simple approach will — by def-

inition of Target-Regions and since target positions

are ﬁxed — result in optimal solutions in full envi-

ronments. The reasons therefore are that i) agent po-

sitions in full environments can also be considered to

be ﬁxed and ii) that agents are anonymous in the qual-

ity function deﬁned in Equation 1. Accordingly, such

a full scenario reduces to a static one, which means

that TR provide optimal partitionings in each iteration

and thus also result in an optimal average partitioning

quality.

Although these regions were calculated for a full

and thus static scenario, we also propose to solve the

general iterative partitioning problem, which usually

involves less agents, with the help of these static re-

gions. Since we then work in so-called sparse envi-

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

202

ronments, Target-Regions by deﬁnition may not pro-

vide optimal solutions in all iterations.

At this point, two questions arise: the ﬁrst ques-

tion concerns the potential of static Target-Regions as

a mean to solve the iterative problem in sparse envi-

ronments. The second question is about the complex-

ity of determining Target-Regions in static scenarios.

We investigate question one using Hypothesis 1.

Hypothesis 1. Let agents in an A-T -Grid-Environ-

ment repeatedly create partitionings based on Target-

Regions. Then the resulting average partitioning qual-

ity according to Equation 2 is expected to be

1. high if agents are distributed uniformly, and

2. low if agents are distributed according to a normal

distribution.

The solution quality further depends on the ratios be-

tween the number of agents, targets, and the size of

the environment.

Note that if agents are normally distributed then they

are concentrated within a certain part of the environ-

ment. This may lead to situations where no or only

a small number of agents is located in some Target-

Regions. Accordingly, bad distribution values and

thus bad overall partitioning qualities result. Hence,

in the remainder of this work we concentrate on uni-

formly distributed agents. In addition, the results

of an empirical analysis fully support Hypothesis 1.

The corresponding experiments compare optimal so-

lutions for settings with two targets to those obtained

using Algorithm 1 and static Target-Regions in sparse

environments.

Details can be found in the extended

paper (Kemmerich and Kleine B

uning, 2010b).

The second question concerning the complexity

of calculating Target-Regions is not yet answered.

As by deﬁnition, Target-Regions are based on opti-

mal solutions, their construction requires to optimally

solve the partitioning problem. However, as already

mentioned we conjecture that solving such static sce-

narios with more than two targets is at least NP-

hard, since no efﬁcient algorithm is known (Goebels,

2007). Thus, the problem of determining Target-

Regions is conjectured to be at least NP-hard, too.

We considered settings with two targets, because no

polynomial-time algorithm that provably returns an optimal so-

lution for settings with an arbitrary number of targets is known

(Goebels, 2007). Accordingly, validating Hypothesis 1 for general

settings is computationally intractable. However, we are aware of

a central-instance polynomial-time algorithm for settings with two

targets which we used in the evaluation.

5 APPROXIMATION OF

TARGET-REGIONS

As the experiments conducted for validating Hypoth-

esis 1 resulted in high quality solutions for uniformly

distributed agents and because the construction of

Target-Regions is assumed to be at least NP-hard, we

propose to use approximated Target-Regions. The

presented approximation is based on a local algorithm

that is known as Exchange Target Strategy (ETS)

(Goebels, 2007). Hence, we call the approximated

regions ETS-Target-Regions.

5.1 ETS-Target-Regions

According to (Goebels, 2007), the Exchange Target

Strategy (ETS) is a good mean to ﬁnd high qual-

ity partitionings of agents to targets in settings with

static positions. The basic idea of the ETS is as fol-

lows. Initially, agents are (randomly) assigned to tar-

gets. Then, agents repeatedly communicate assign-

ment and distance information. They exchange target

assignments with neighboring agents if this locally

improves the distance objective. Thus, the distribu-

tion objective is ﬁxed based on the initial assignment

while the distance objective gradually improves until

it converges. More details on ETS can be found in the

extended version of the paper or in (Goebels, 2007).

Although ETS provides high quality solutions on

average, worst cases leading to poor solutions or lo-

cal optima can be constructed (Goebels, 2007). In

addition, the costs produced by repeated information

exchange may become relatively high.

To approximate Target-Regions, we propose to

use the Exchange Target Strategy (ETS). We deﬁne

the resulting regions in Deﬁnition 3.

Deﬁnition 3 (ETS-Target-Region)

An ETS-Target-Region ETS-TR(t) for any target t in

a full A-T -Grid-Environment is deﬁned by a set of

cells that consists of target t’s cell and all cells whose

agents are assigned to t after the Exchange Target

Strategy has converged.

Note that ETS-Target-Regions (ETS-TR) in this work

are those that have evolved after 2000 iterations of the

ETS approach, as hand-made experiments indicated

that this value was by far sufﬁcient for convergence

in all considered scenarios. Convergence in this con-

text means that no further improvement of the overall

solution quality was observed after 2000 iterations.

We developed an approach that ﬁrst calculates

ETS-Target-Regions for a full environment. The re-

sulting ETS-TR then are mapped to the cells of the

REGION-BASED HEURISTICS FOR AN ITERATIVE PARTITIONING PROBLEM IN MULTIAGENT SYSTEMS

203

grid environment, i.e. each cell obtains an informa-

tion about the region it belongs to. In a second phase,

this initialized environment then can be used with the

desired number of agents. Each agent simply selects

the target that is identiﬁed by the information stored

at the agent’s current position.

Again, we simulated this approach and compared

the results to optimal solutions for settings with two

targets. We calculated an average error, which we de-

ﬁned as the average difference between the optimal

solution value and the solution value obtained using

the ETS-Target-Regions over all iterations. We found

that ETS-TR with an appropriate ratio between the

number of agents and the environment size is roughly

1% or less in seven out of nine simulated scenar-

ios. Accordingly, we can conclude that ETS-TR are a

good mean to approach the iterative partitioning prob-

lem, too. A more detailed description and the experi-

mental evaluation can be found in the extended paper

(Kemmerich and Kleine B

uning, 2010b).

5.2 Sparse-ETS-Target-Region

Algorithm

As indicated in Section 5.1, ETS-TR are a promising

mean to solve our iterative partitioning task. How-

ever, it is impractical to consider a full A-T -Grid-

Environment or to apply an approach that calculates

ETS-Target-Regions to initialize the cells. In this sec-

tion we thus introduce a local algorithm for sparse

environments that approximates ETS-Target-Regions.

Therefore, we use so-called storage media for coordi-

nation (Kemmerich and Kleine B

uning, 2010a). Stor-

age media (SM) are located at ﬁxed positions in the

environment and can be used by agents to store infor-

mation externally.

The Sparse ETS-Target-Region (S-ETS-TR) ap-

proach can basically be divided into two phases. In

the ﬁrst phase, each agent locally executes the ETS

algorithm for a ﬁxed number of iterations i

max

. Af-

ter each ETS iteration, every agent a stores its current

target assignment combined with its current position

pos(a) on a storage media in its vicinity. Therefore,

it must be guaranteed that each agent can always in-

teract with a storage media. Then, the combination

of all information stored on all storage media from

a global perspective represents approximated ETS-

Target-Regions, as sketched in Figure 2.

In the second phase agents stop to execute the ETS

algorithm. Instead, they retrieve a target assignment

from the nearest SM based on the information stored

at the media. These information approximate ETS-

Target-Regions. Depending on the parameters, there

will be positions without target assignment informa-

Target 1

Target 2

Target 3

Storage medium

approximated

ETS-Target-Region

Figure 2: Exemplary target assignment information (col-

ored boxes) stored on the storage media after some S-ETS-

TR iterations.

tion (white boxes in Figure 2). If this happens, a given

position must be classiﬁed according to locally avail-

able information. Therefore, a SM executes a simple

classiﬁcation algorithm that uses a counting argument

based on eight surrounding positions. The resulting

classiﬁcation basically corresponds to the most often

selected target in the surrounding. It is returned and

stored on the storage media for later use.

Algorithm 2 summarizes the S-ETS-TR approach.

Details on the data structures and on the classiﬁcation

algorithm as well as on the experimental results are

given in the extended paper.

Algorithm 2: Executed by each agent a ∈ A.

1: procedure SPARSE-ETS-TARGET-REGIONS

2: m ← nearest SM

3: if current iteration < i

max

then  Phase 1

4: t ← target assignment of an ETS iteration

5: m.STORE-INFO(t, pos(a))

6: else  Phase 2

7: t ← m.CLASSIFY-POSITION(pos(a))

8: if t is set then assign agent to t

9: else keep last target assignment

Figure 3 brieﬂy summarize the latter results. It

illustrates the process of both ETS-based approaches

and shows 95% conﬁdence intervals. The vertical line

at iteration 500 marks the beginning of the second

phase. It is particularly noticeable that phase one of

S-ETS-TR may produce higher solution values as op-

timization is performed in each iteration by executing

the ETS algorithm. Performance decreases in the sec-

ond phase depending on the settings, esp. in settings

with a higher target to agent ratio. However, with a

lower ratio, the results in the second phase become

even better (e.g. for 5 targets and 500 agents).

Comparing the average partitioning quality q, we

observe that the local S-ETS-TR heuristic is able to

produce high quality solutions that are on a level with

the real ETS-TR approach from Section 5.1. This is

particularly interesting as the very simple classiﬁca-

tion technique misclassiﬁes between 5% and 23% of

the cells compared to real ETS-Target-Regions.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

204

0 200 400 600 800 1000

0.0 0.2 0.4 0.6 0.8 1.0

Iteration

solution quality

●

2 targets, 100 agents (q=0.97646)

2 targets, 500 agents (q=0.98113)

5 targets, 100 agents (q=0.88665)

5 targets, 500 agents (q=0.92793)

16 targets, 100 agents (q=0.52215)

16 targets, 500 agents (q=0.73970)

●

(a) S-ETS-TR

0 200 400 600 800 1000

0.0 0.2 0.4 0.6 0.8 1.0

Iteration

solution quality

●

● ●

●

2 targets, 100 agents (q=0.97611)

2 targets, 500 agents (q=0.98028)

5 targets, 100 agents (q=0.89079)

5 targets, 500 agents (q=0.93000)

16 targets, 100 agents (q=0.54179)

16 targets, 500 agents (q=0.80675)

●

(b) ETS-TR

Figure 3: Solution qualities with 95% conﬁdence intervals.

The results of the ETS-TR approach in Figure

3(b) also illustrate the potential of ETS-based Target-

Region approximations in settings with many targets.

However, they are also affected by a dependency on

the ratio between the number of agents and targets as

already stated for Target-Regions in Hypothesis 1.

6 CONCLUSIONS

We constructed a local region-based heuristic to solve

an iterative partition task in a multiagent system. The

potential of the general idea and different region types

were investigated experimentally. Our simulation re-

sults, although more simulations in different settings

should be performed in the future, attribute high po-

tential to region based heuristics. To improve solu-

tion qualities of the local approach, other classiﬁca-

tion techniques should be investigated to reach the

potential that approximated regions can provide. Re-

turning to our intension of building systems that are

able to solve the iterative partitioning problem with

very basic and simple information structures and rea-

soning processes, region-based approaches provide a

promising mean and should be investigated further.

The extended version of this work (Kemmerich

and Kleine B

uning, 2010b) provides additional details

on the approaches and their evaluation and proves

some properties of optimal solutions and Target-

Regions. We also present a formula to estimate so-

lution qualities of the ETS-TR approach and discuss

some cost issues concerning the local heuristic.

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