Algorithms and Ordering Heuristics for Distributed Constraint Satisfaction Problems

166 Pages·2013·4.998 MB·English

by  Mohamed Wahbi

Algorithms and Ordering Heuristics for Distributed
Constraint Satisfaction Problems
In memoryof a great man,
Mouhamed Moumane
FOCUS SERIES
Series Editor Narendra Jussien
Algorithms and Ordering
Heuristics for Distributed
Constraint Satisfaction
Problems
Mohamed Wahbi
Firstpublished2013 inGreatBritainandtheUnitedStatesbyISTELtdandJohnWiley&Sons,Inc.
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Contents
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
PART1.BACKGROUNDONCENTRALIZEDAND
DISTRIBUTEDCONSTRAINTREASONING . . . . . . . . . . . . . . . . . . . 1
CHAPTER1.CONSTRAINTSATISFACTION PROBLEMS . . . . . . . . . . . 3
1.1.Centralizedconstraintsatisfactionproblems. . . . . . . . . . . . . . . . 3
1.1.1.Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2.ExamplesofCSPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.AlgorithmsandtechniquesforsolvingcentralizedCSPs. . . . . . . . . 10
1.2.1.AlgorithmsforsolvingcentralizedCSPs . . . . . . . . . . . . . . . 10
1.2.2.VariableorderingheuristicsforcentralizedCSPs . . . . . . . . . . . 23
1.3.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
CHAPTER2.DISTRIBUTEDCONSTRAINTSATISFACTION PROBLEMS . . 29
2.1.Distributedconstraintsatisfaction
problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1.Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2.ExamplesofDisCSPs . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.3.Distributedmeetingschedulingproblem(DisMSP) . . . . . . . . . 32
2.1.4.Distributedsensornetworkproblem
(SensorDCSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.MethodsforsolvingDisCSPs . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1.SynchronoussearchalgorithmsonDisCSPs . . . . . . . . . . . . . 37
2.2.2.AsynchronoussearchalgorithmsonDisCSPs. . . . . . . . . . . . . 40
vi AlgorithmsandOrderingHeuristicsforDisCSPs
2.2.3.DynamicorderingheuristicsonDisCSPs . . . . . . . . . . . . . . . 44
2.2.4.MaintainingarcconsistencyonDisCSPs . . . . . . . . . . . . . . . 47
2.3.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
PART2.SYNCHRONOUSSEARCHALGORITHMSFOR DISCSPS . . . . . 49
CHAPTER3.NOGOOD-BASED ASYNCHRONOUSFORWARD
CHECKING(AFC-NG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.Nogood-basedasynchronousforwardchecking . . . . . . . . . . . . . . 53
3.2.1.Descriptionofthealgorithm . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2.Asimpleexampleofthebacktrackoperation
onAFC-likealgorithms . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.Correctnessproofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.Experimentalevaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.1.UniformbinaryrandomDisCSPs . . . . . . . . . . . . . . . . . . . 61
3.4.2.Distributedsensor-targetproblems . . . . . . . . . . . . . . . . . . . 62
3.4.3.Distributedmeetingschedulingproblems . . . . . . . . . . . . . . . 64
3.4.4.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
CHAPTER4.ASYNCHRONOUSFORWARD-CHECKING
TREE (AFC-TREE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.Pseudo-treeordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.Distributeddepth-firstsearchtreeconstruction . . . . . . . . . . . . . . 72
4.4.TheAFC-treealgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.1.Descriptionofthealgorithm . . . . . . . . . . . . . . . . . . . . . . 77
4.5.Correctnessproofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6.Experimentalevaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6.1.UniformbinaryrandomDisCSPs . . . . . . . . . . . . . . . . . . . 80
4.6.2.Distributedsensor-targetproblems . . . . . . . . . . . . . . . . . . . 82
4.6.3.Distributedmeetingschedulingproblems . . . . . . . . . . . . . . . 84
4.6.4.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7.Otherrelatedworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.8.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
CHAPTER5.MAINTAININGARCCONSISTENCY ASYNCHRONOUSLY
INSYNCHRONOUSDISTRIBUTEDSEARCH . . . . . . . . . . . . . . . . . . 87
5.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.Maintainingarcconsistency. . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.Maintainingarcconsistencyasynchronously . . . . . . . . . . . . . . . 89
5.3.1.EnforcingACusingdelmessages(MACA-del) . . . . . . . . . . . 90
Contents vii
5.3.2.EnforcingACwithoutadditionalkindofmessage
(MACA-not) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.Theoreticalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5.Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5.1.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
PART3.ASYNCHRONOUSSEARCHALGORITHMSAND
ORDERINGHEURISTICSFORDISCSPS . . . . . . . . . . . . . . . . . . . . 101
CHAPTER6.CORRIGENDUMTO“MIN-DOMAINRETROACTIVE
ORDERING FORASYNCHRONOUSBACKTRACKING” . . . . . . . . . . . . 103
6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3.ABT_DO-Retromaynotterminate . . . . . . . . . . . . . . . . . . . . . 106
6.4.Therightwaytocompareorders . . . . . . . . . . . . . . . . . . . . . . 108
6.5.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
CHAPTER7.AGILEASYNCHRONOUSBACKTRACKING
(AGILE-ABT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2.Introductorymaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2.1.Reorderingdetails . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2.2.Thebacktrackingtarget . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.2.3.Decreasingterminationvalues . . . . . . . . . . . . . . . . . . . . . 116
7.3.Thealgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.4.Correctnessandcomplexity . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.5.Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5.1.UniformbinaryrandomDisCSPs . . . . . . . . . . . . . . . . . . . 124
7.5.2.Distributedsensortargetproblems . . . . . . . . . . . . . . . . . . . 125
7.5.3.Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.6.Relatedworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.7.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
PART4.DISCHOCO2.0: APLATFORMFORDISTRIBUTED
CONSTRAINTREASONING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
CHAPTER8.DISCHOCO2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2.Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.1.Communicationsystem . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.2.Eventmanagement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.2.3.Observersinlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
viii AlgorithmsandOrderingHeuristicsforDisCSPs
8.3.UsingDisChoco2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4.Experimentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.5.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Preface
Constraintprogrammingisanareaincomputersciencethathasgainedincreasing
interestinrecentyears.Constraintprogrammingisbasedonitspowerfulframework
called constraint satisfaction problem (CSP). CSP is a general framework that can
formalize many real-world combinatorial problems such as resource allocation, car
sequencing, natural language understanding and machine vision. A CSP consists of
looking for solutions to a constraint network, i.e. a set of assignments of values to
variables that satisfy the constraints of the problem. These constraints represent
restrictionsonvaluecombinationsallowedforconstrainedvariables.
Various applications that are of a distributed nature exist. In this kind of
application, the knowledge about the problem, i.e. variables and constraints, is
distributed among physically distributed agents. This distribution is mainly due to
privacyand/orsecurityrequirements: constraintsorpossiblevaluesmaybestrategic
information that should not be revealed to other agents that can be seen as
competitors. Several applications in multi-agent coordination are of such kind.
Examplesofapplicationsaresensornetworks[JUN01,BÉJ05],militaryunmanned
aerialvehicleteams[JUN01],distributedschedulingproblems[WAL02,MAH04],
distributed resource allocation problems [PET04], log-based reconciliation
[CHO06], distributed vehicle routing problems [LÉA11], etc. Therefore, the
distributedframeworkdistributedconstraintsatisfactionproblem(DisCSP)isusedto
modelandsolvethiskindofproblem.
A DisCSP is composed of a group of autonomous agents, where each agent has
controlofsomeelementsofinformationaboutthewholeproblem, i.e.variablesand
constraints. Each agent owns its local constraint network. Variables in different
agents are connected by constraints. Agents must assign, in a distributed manner,
values to their variables so that all constraints are satisfied. Hence, agents assign
values to their variables, attempting to generate locally consistent assignments that
arealsoconsistentwithconstraintsbetweenagents[YOK98,YOK00a].Toachieve
thisgoal,agentscheckthevaluesassignedtotheirvariablesforlocalconsistencyand
x AlgorithmsandOrderingHeuristicsforDisCSPs
exchange messages to check the consistency of their proposed assignments against
constraintsthatcontainvariablesthatbelongtootheragents.
Many distributed algorithms for solving DisCSPs have been designed in the last
two decades. They can be divided into two main groups: synchronous and
asynchronous algorithms. The first category includes algorithms in which agents
assign values to their variables in a synchronous and sequential way. The second
category includes algorithms in which the process of proposing values to the
variables and exchanging these proposals is performed asynchronously between the
agents. In the former category, agents do not have to wait for decisions of others,
whereas, in general, only one agent has the privilege of making a decision in the
synchronousalgorithms.
Thisbooktriestoextendthestateoftheartbyproposingseveralalgorithmsand
heuristicsforsolvingtheDisCSPs.Thebookstartswithabriefintroductiontothestate
oftheartintheareaofcentralizedconstraintprogramming.The(CSP)formalismis
definedandsomeacademicandrealexamplesofproblemsthatcanbemodeledand
solvedbyCSParepresented.Then,typicalmethodsforsolvingcentralizedCSPsare
briefly reported. Next, preliminary definitions on the DisCSP formalism are given.
Afterward, the main algorithms that have been developed in the literature to solve
DisCSPsaredescribed.
The second part of this book provides three algorithms for solving DisCSPs.
These algorithms are classified under the category of synchronous algorithms. The
first algorithm is the nogood-based asynchronous forward checking (AFC-ng).
AFC-ng is a nogood-based version of the asynchronous forward checking (AFC)
[MEI07] algorithm. Besides its use of nogoods as justification of value removals,
AFC-ng allows simultaneous backtracks to go from different agents to different
destinations. AFC-ng only needs polynomial space. Proofs of the correctness of the
AFC-ng are also given. A comparison of its performance with other well-known
distributed algorithms for solving DisCSP is presented. The results are reported for
random DisCSPs and instances from real benchmarks: sensor networks and
distributedmeetingscheduling.
The second algorithm, called asynchronous forward-checking tree (AFC-tree),
extends the AFC-ng algorithm using a pseudo-tree arrangement of the constraint
graph. To achieve this goal, agents are ordered a priori in a pseudo-tree such that
agentsindifferentbranchesofthetreedonotshareanyconstraint.AFC-treedoesnot
address the process of ordering the agents in a pseudo-tree arrangement. The
construction of the pseudo-tree is done in a preprocessing step. Using this priority
ordering, AFC-tree performs multiple AFC-ng processes on the paths from the root
to the leaves of the pseudo-tree. The good properties of the AFC-tree are
demonstrated. AFC-tree is compared to AFC-ng on random DisCSPs and instances
fromrealbenchmarks: sensornetworksanddistributedmeetingscheduling.