Lecture Notes in Artificial Intelligence 5883

Edited by R. Goebel, J. Siekmann, and W. Wahlster

Subseries of Lecture Notes in Computer Science

Roberto Serra Rita Cucchiara (Eds.)

AI*IA 2009:Emergent Perspectivesin Artificial Intelligence

XIth International Conference of the

Italian Association for Artificial Intelligence

Reggio Emilia, Italy, December 9-12, 2009

Proceedings

13

Series Editors

Randy Goebel, University of Alberta, Edmonton, CanadaJörg Siekmann, University of Saarland, Saarbrücken, GermanyWolfgang Wahlster, DFKI and University of Saarland, Saarbrücken, Germany

Volume Editors

Roberto SerraUniversità di Modena e Reggio EmiliaDipartimento di Scienze Sociali, Cognitive e QuantitativeVia Allegri 9, 42121 Reggio Emilia, ItaliaE-mail: roberto.serra@unimore.it

Rita CucchiaraUniversità di Modena e Reggio EmiliaDipartimento di Ingegneria dell‘InformazioneVia Vignolese 905, 41100 Modena, ItaliaE-mail: rita.cucchiara@unimore.it

Library of Congress Control Number: 2009939107

CR Subject Classification (1998): I.2, F.1, F.4, I.2.6, H.2.8, I.5

LNCS Sublibrary: SL 7 – Artificial Intelligence

ISSN 0302-9743

ISBN-10 3-642-10290-5 Springer Berlin Heidelberg New York

ISBN-13 978-3-642-10290-5 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,in its current version, and permission for use must always be obtained from Springer. Violations are liableto prosecution under the German Copyright Law.

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© Springer-Verlag Berlin Heidelberg 2009Printed in Germany

Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, IndiaPrinted on acid-free paper SPIN: 12786985 06/3180 5 4 3 2 1 0

Preface

This volume contains the scientific papers accepted for publication at the confer-ence of the Italian Association for Artificial Intelligence (AI*IA), held in ReggioEmilia during December 9-12, 2009. This was the 11th conference of the series,whose previous editions have been held every two years since 1989, in Trento,Palermo, Turin, Florence, Rome, Bologna, Bari, Pisa, Milan and Rome again.

The conference’s scope is broad, and covers all the aspects of artificial intel-ligence. Every edition has, however, an individual flavor, and in this case, bothby the choice of some invited speakers and of a dedicated workshop, we stressedthe growing importance of complex systems methods and ideas in AI.

Eighty-two papers were submitted from 13 different countries located in fourcontinents, and the Program Committee selected the 50 papers which are pub-lished in this volume. I wish to thank all the members of the Program Committeeand the other reviewers for their excellent job.

The three invited speakers, who provided a fundamental contribution to thesuccess of the conference, were Shay Bushinsky (Haifa), Marco Dorigo (Brussels)and Paul Verschure (Barcelona).

The conference also hosted a number of interesting workshops, which raisedconsiderable interest and participation:

“Bio-logical: Logic-Based Approaches in Bioinformatics,” organized by StefanoFerilli and Donato Malerba.“Complexity, Evolution and Emergent Intelligence,” organized by Marco Villaniand Stefano Cagnoni.“Evalita 2009: Workshop on Evaluation of NLP and Speech Tools for Italian,”organized by Bernardo Magnini and Amedeo Cappelli.“Intelligent Cultural Heritage,” organized by Luciana Bordoni.“Intelligenza Artificiale ed e-Learning,” organized by Giovanni Adorni.“Pattern Recognition and Artificial Intelligence for Human Behavior Analysis,”organized by Luca Iocchi, Andrea Prati and Roberto Vezzani.“RCRA 2009: Experimental Evaluation of Algorithms for Solving Problems withCombinatorial Explosion,” organized by Marco Gavanelli and Toni Mancini.

Special thanks are due to the members of the local Organizing Committee, whocarried a heavy organizational burden. I am particularly indebted to my Co-chairRita Cucchiara, who shared the task of defining all the main choices concerningthe conference, and to Andrea Prati, who was of great help in organizing thepaper-review process. My colleague and friend Marco Villani deserves specialgratitude as he continuously helped in all the issues related to the conferenceorganization with care and competence.

VI preface

I am also grateful to the members of the local organizing team, my PhDstudents Alex Graudenzi and Chiara Damiani, later joined by Alessia Barbieriand Luca Ansaloni, who were essential in innumerable aspects, including (butcertainly not limited to) the preparation and update of the conference website,the preparation of the camera-ready proceedings volume and the handling of thecorrespondence.

The conference was hosted in the beautiful palace were the Faculty of Com-munications and Economics is located; I thank the Faculty and its dean for theavailability of these spaces and conference rooms, which required re-scheduling ofsome lessons. The University of Modena and Reggio Emilia and the Departmentof Social, Cognitive and Quantitative Sciences are also gratefully acknowledgedfor their patronage. I also thank Anna Kramer and the staff of Springer’s LectureNotes in Artificial Intelligence for their timely and effective editorial work.

A conference like this is also a chance for letting the discipline of AI becomebetter known to laymen and to the general public; several satellite initiativesof this kind were launched, and I thank Shay Bushinsky, Marco Gori, DanieleNardi and Gianni Zanarini for their support, and the municipality of ReggioEmilia for financial support.

Finally, I am glad to acknowledge the essential role of AI*IA and of its Boardof Directors, and in particular of its president Marco Schaerf, who was very activein all the phases of the preparation of this event. I also received invaluable sup-port and advice from my colleagues, the other ex-presidents of the association:Luigia Carlucci Aiello, Oliviero Stock, Pietro Torasso, Marco Gori.

September 2009 Reggio EmiliaRoberto Serra

Organization

AI*IA 2009 was organized by the Italian Association for Artificial Intelligence(AI*IA) in cooperation with the department of Social, Cognitive and Quantita-tive Sciences and the department of Information Engineering of the Universtyof Modena and Reggio Emilia

Organizing Committee

Roberto Serra (Modena and Reggio Emilia) Conference ChairRita Cucchiara (Modena and Reggio Emilia) Co-chairFranco Zambonelli (Modena and Reggio Emilia) Local Events ChairMarco Villani (Modena and Reggio Emilia) Workshop CoordinatorAndrea Prati (Modena and Reggio Emilia) Refereeing CoordinatorMarco Mamei (Modena and Reggio Emilia) Local Arrangements

Coordinator

Program Committee

Khurshid AhmadLuigia Carlucci AielloStefania BandiniRoberto BasiliErnesto BurattiniStefano CagnoniMarc CavazzaRoberto CordeschiRita CucchiaraWalter DaelemansFloriana EspositoSalvatore GaglioMarco GavanelliAttilio GiordanaMarco GoriNicola GuarinoEvelina LammaMarco Mamei

Toni ManciniPaola MelloAndrea OmiciniPatrizia PaggioTeresa PazienzaAndrea PratiAndrea RoliRoberto SerraMarco ShaerfKiril SimovGiovanni SodaSteffen StaabOliviero StockTokunaga TakenobuPietro TorassoMarco VillaniRoman YangarberFranco Zambonelli

Local Arrangements

Alex GraudenziChiara Damiani

VIII Organization

Luca AnsaloniAlessia Barbieri(Modena and Reggio Emilia University, Italy)

Referees

K. AhmadL. Carlucci AielloM. AndrettaB. ApolloniS. BandiniR. BasiliA. BorgheseM. BorrottiE. BurattiniR. CampaniniS. CagnoniM. CavazzaR. CordeschiR. CucchiaraW. Daelemans

F. EspositoS. GaglioM. GavanelliA. GiordanaM. GoriN. GuarinoE. LammaP. LiberatoreM. MameiT. ManciniP. MelloM. MilanoM. MirolliA. OmiciniP. Paggio

T. PazienzaA. PratiA. RoliR. SerraM. ShaerfK. SimovG. SodaS. StaabO. StockT. TakenobuP. TorassoM. VillaniR. YangarberF. Zambonelli

Sponsoring Institutions

AI*IA (Italian Association for Artificial Intelligence) and University of Modenaand Reggio Emilia.

Table of Contents

Knowledge Representation and Reasoning

Bayesian Networks: The Parental Synergy and the Prior ConvergenceError . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Janneke H. Bolt

Constraints for Representing Transforming Entities in Bio-ontologies . . . 11C. Maria Keet

Common-Sense Rule Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Ilaria Lombardi and Luca Console

Hard QBF Encodings Made Easy: Dream or Reality? . . . . . . . . . . . . . . . . 31Luca Pulina and Armando Tacchella

ONGAS: A COllaborative Ontology Development Framework Based onNamed GrAphS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Daniele Bagni, Marco Cappella, Maria Teresa Pazienza, andArmando Stellato

Pstable Semantics for Logic Programs with Possibilistic OrderedDisjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Roberto Confalonieri, Juan Carlos Nieves, andJavier Vazquez-Salceda

Reasoning about Typicality with Low Complexity Description Logics:The Logic EL+⊥

T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Laura Giordano, Valentina Gliozzi, Nicola Olivetti, andGian Luca Pozzato

A Prioritized “And” Aggregation Operator for MultidimensionalRelevance Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Celia da Costa Pereira, Mauro Dragoni, and Gabriella Pasi

Deriving Information from Sampling and Diving . . . . . . . . . . . . . . . . . . . . . 82Michele Lombardi, Michela Milano, Andrea Roli, andAlessandro Zanarini

Optimal Decision Tree Synthesis for Efficient NeighborhoodComputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Costantino Grana and Daniele Borghesani

Mathematical Symbol Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Simone Marinai, Beatrice Miotti, and Giovanni Soda

C

X Table of Contents

Machine Learning

Local Kernel for Brains Classification in Schizophrenia . . . . . . . . . . . . . . . . 112U. Castellani, E. Rossato, V. Murino, M. Bellani, G. Rambaldelli,M. Tansella, and P. Brambilla

Improvement of the Classifier Performance of a Pedestrian DetectionSystem by Pixel-Based Data Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Holger Lietz, Jan Thomanek, Basel Fardi, and Gerd Wanielik

Plugging Taxonomic Similarity in First-Order Logic Horn ClausesComparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

S. Ferilli, M. Biba, N. Di Mauro, T.M.A. Basile, and F. Esposito

Empirical Assessment of Two Strategies for Optimizing the ViterbiAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Roberto Esposito and Daniele P. Radicioni

Approximate Frequent Itemset Discovery from Data Stream . . . . . . . . . . . 151Anna Ciampi, Fabio Fumarola, Annalisa Appice, andDonato Malerba

Kernel-Based Learning for Domain-Specific Relation Extraction . . . . . . . . 161Roberto Basili, Cristina Giannone, Chiara Del Vescovo,Alessandro Moschitti, and Paolo Naggar

Automatic Cluster Selection Using Index Driven Search Strategy . . . . . . . 172Denis Ferraretti, Giacomo Gamberoni, and Evelina Lamma

The Necessity of Machine Learning and Epistemology in theDevelopment of Categorization Theories: A Case Study inPrototype-Exemplar Debate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Francesco Gagliardi

Evolutionary Computation

A Lexicographic Encoding for Word Sense Disambiguation withEvolutionary Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

A. Azzini, C. da Costa Pereira, M. Dragoni, and A.G.B. Tettamanzi

Feature Membership Functions in Voronoi-Based Zoning . . . . . . . . . . . . . . 202S. Impedovo, A. Ferrante, R. Modugno, and G. Pirlo

Optimal Planning with ACO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212M. Baioletti, A. Milani, V. Poggioni, and F. Rossi

Navigation in Evolving Robots: Insight from Vertebrates. The Case ofGeometric and Non-geometric Information . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Michela Ponticorvo and Orazio Miglino

Table of Contents XI

Search

Social Tagging for Personalized Web Search . . . . . . . . . . . . . . . . . . . . . . . . . 232Claudio Biancalana

Partitioning Search Spaces of a Randomized Search . . . . . . . . . . . . . . . . . . 243Antti E.J. Hyvarinen, Tommi Junttila, and Ilkka Niemela

Improving Plan Quality in SAT-Based Planning . . . . . . . . . . . . . . . . . . . . . 253Enrico Giunchiglia and Marco Maratea

A New Approach to Iterative Deepening Multiobjective A* . . . . . . . . . . . . 264J. Coego, L. Mandow, and J.L. Perez de la Cruz

High Performing Algorithms for MAP and Conditional Inference inMarkov Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Marenglen Biba, Stefano Ferilli, and Floriana Esposito

Natural Language Processing

A Robust Geometric Model for Argument Classification . . . . . . . . . . . . . . . 284Cristina Giannone, Danilo Croce, Roberto Basili, and Diego De Cao

Probabilistic Ontology Learner in Semantic Turkey . . . . . . . . . . . . . . . . . . . 294Francesca Fallucchi, Noemi Scarpato, Armando Stellato, andFabio Massimo Zanzotto

Semantic Annotation of Legal Modificatory Provisions . . . . . . . . . . . . . . . . 304Leonardo Lesmo, Alessandro Mazzei, and Daniele P. Radicioni

Towards Extensible Textual Entailment Engines: The EDITSPackage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Matteo Negri, Milen Kouylekov, Bernardo Magnini,Yashar Mehdad, and Elena Cabrio

“Language Is the Skin of My Thought”: Integrating Wikipedia and AIto Support a Guillotine Player . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Pasquale Lops, Pierpaolo Basile, Marco de Gemmis, andGiovanni Semeraro

Analyzing Interactive QA Dialogues Using Logistic RegressionModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Manuel Kirschner, Raffaella Bernardi, Marco Baroni, andLe Thanh Dinh

XII Table of Contents

Multi-agent Systems

An Agent-Based Paradigm for Free-Hand Sketch Recognition . . . . . . . . . . 345D.G. Fernandez-Pacheco, J. Conesa, N. Aleixos, P. Company, andM. Contero

Solving Integer Programming Problems by Using Artificial Bee ColonyAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Bahriye Akay and Dervis Karaboga

Representing Excuses in Social Dependence Networks . . . . . . . . . . . . . . . . . 365Guido Boella, Jan Broersen, Leendert van der Torre, andSerena Villata

Application

Statistical and Fuzzy Approaches for Atmospheric Boundary LayerClassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Angelo Ciaramella, Angelo Riccio, Federico Angelini,Gian Paolo Gobbi, and Tony Christian Landi

Learning Local Correspondences for Static Signature Verification . . . . . . . 385G. Pirlo, D. Impedovo, E. Stasolla, and C.A. Trullo

ML Techniques for the Classification of Car-Following Maneuver . . . . . . . 395Fabio Tango and Marco Botta

An Asynchronous Cellular Automata-Based Adaptive IlluminationFacility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Stefania Bandini, Andrea Bonomi, Giuseppe Vizzari, andVito Acconci

Relational Temporal Data Mining for Wireless Sensor Networks . . . . . . . . 416Teresa M.A. Basile, Nicola Di Mauro, Stefano Ferilli, andFloriana Esposito

Ontology-Driven Co-clustering of Gene Expression Data . . . . . . . . . . . . . . 426Francesca Cordero, Ruggero G. Pensa, Alessia Visconti,Dino Ienco, and Marco Botta

Value-Driven Characters for Storytelling and Drama . . . . . . . . . . . . . . . . . . 436Rossana Damiano and Vincenzo Lombardo

A Knowledge Management System Using Bayesian Networks . . . . . . . . . . 446Patrizia Ribino, Antonio Oliveri, Giuseppe Lo Re, andSalvatore Gaglio

Knowledge Discovery and Digital Cartography for the ALS (LinguisticAtlas of Sicily) Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Antonio Gentile, Roberto Pirrone, and Giuseppe Russo

Table of Contents XIII

Parameter Tuning of a Stochastic Biological Simulator byMetaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

Sara Montagna and Andrea Roli

Hypotheses about Typical General Human Strategic Behavior in aConcrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Rustam Tagiew

A Fuzzy Approach to Product Configuration on Standard Databases . . . 486Luigi Portinale

Classifying Human Body Acceleration Patterns Using a HierarchicalTemporal Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

Federico Sassi, Luca Ascari, and Stefano Cagnoni

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

A New Approach to Iterative DeepeningMultiobjective A*

J. Coego, L. Mandow, and J.L. Pérez de la Cruz�

Dpto. Lenguajes y Ciencias de la Computación, Universidad de Málaga 29071,Málaga, Spain

{jcoego,lawrence,perez}@lcc.uma.es

Abstract. Multiobjective search is a generalization of the Shortest PathProblem where several (usually conflicting) criteria are optimized simul-taneously. The paper presents an extension of the single-objective IDA*search algorithm to the multiobjective case. The new algorithm is il-lustrated with an example, and formal proofs are presented on its ter-mination, completeness, and admissibility. The algorithm is evaluatedover a set of random tree search problems, and is found to be more effi-cient than IDMOA*, a previous extension of IDA* to the multiobjectivecase.

1 Introduction

Heuristic search in Shortest Path Problems is a central field of study in Ar-tificial Intelligence. The Multiobjective Shortest Path Problem (MSPP) is anextension of the Shortest Path Problem with practical applications in differentdomains, like circuit partitioning [1], operator scheduling, channel routing [2], ordomain independent planning [3]. Multiobjective problems require the evalua-tion of several different and frequently conflicting objectives for each alternative.These problems rarely have a single optimal solution. Most frequently, a set ofnon-dominated (Pareto-optimal) solutions can be found, each one presenting aparticular trade-off between the objectives under consideration. The number ofnon-dominated solutions in MSPP is known to grow exponentially with solutiondepth in the worst case [4]. Fortunately, several classes of interesting multiob-jective problems do not exhibit this worst-case behavior [5].

This paper deals with depth-first search strategies, which present the advan-tage of memory requirements linear with the depth of the solution. However, incertain cases these algorithms may involve the consideration of an exponentiallylarger set of paths when compared to best-first algorithms. Algorithms IDA* [6],and RBFS [7] are members of this class. Previous multiobjective depth-first al-gorithms include IDMOA* [8], and MOMA*0 [2], the multiobjective extensionsof IDA* and RBFS respectively. This paper presents a natural extension of the� Work partially funded by/Trabajo parcialmente financiado por: Consejería de Inno-

vación, Ciencia y Empresa. Junta de Andalucía (España) - P07-TIC-03018.

R. Serra and R. Cucchiara (Eds.): AI*IA 2009, LNAI 5883, pp. 264–273, 2009.c� Springer-Verlag Berlin Heidelberg 2009

A New Approach to Iterative Deepening Multiobjective A* 265

IDA* algorithm to the multiobjective case. The new algorithm is illustrated withan example, and formal proofs on its termination, completeness and admissibil-ity are presented. Experimental tests over a set of random trees are presented.The algorithm is found to perform more efficiently than IDMOA*, a previousextension of IDA* to the multiobjective case.

The structure of the paper is as follows. Section 2 introduces some commonterminology useful to understand multiobjective problems. Section 3 describesthe new algorithm, and presents an illustrative example and several importantformal properties. Section 4 describes the experimental setup, and analyzes thetime requirements of the algorithms considered. Finally, some conclusions andfuture work are outlined.

2 Multiobjective Shortest Path Problems

Let us consider two q-dimensional vectors v,v� ∈ Rq. A partial order relation ≺denominated dominance is defined as follows, v ≺ v� iff ∀i(1 ≤ i ≤ q) vi ≤ v�iand v �= v�. Given two q-dimensional vectors v and v� (where q > 1), it isnot always possible to say that one is better than the other. For example in abidimensional cost space vector (2, 3) dominates (2, 4), but no dominance relationexists between (2, 3) and (3, 2). Following this, an indifference relation (v ∼ v�)is defined as v neither dominates, nor is dominated by, v�.

Given a set of vectors X , we shall define nd(X) as the set of non-dominatedvectors in X , i. e., nd(X) = {x ∈ X | �y ∈ X y ≺ x} Let G be a locally finitelabeled directed graph G = (N,A, c) with N nodes and A arcs (n, n�) labeledwith positive vectors c(n, n�) ∈ Rq, where c(n, n�) = (c1, . . . , cq) being ci thecost associated to the ith objective. The cost of a path is defined as the sumof the costs of its arcs; obviously, this cost is a q-dimensional vector. Let g(n)denote the cost of the path stored in the search tree from the start node to n,H(n) the set of non-dominated heuristic vectors of node n that estimate the costof a solution from node n to a goal node and F(n) the set of non-dominatedevaluations of node n, computed as f (n) = h(n) + g(n), where h(n) ∈ H(n). Amultiobjective search problem in G is stated as follows:

Given a start node s ∈ N and a set of goal nodes Γ ⊆ N , find the set of allnon-dominated paths P in G, i. e., the set of all paths P such that (i) P goesfrom s to a node in Γ ; (ii) the cost of P is non dominated by the costs of anyother path satisfying (i). Such set is called Γ ∗ and C∗ is the set of all costs ofnon-dominated solution paths.

3 Algorithm PIDMOA∗

Iterative deepening search proceeds by a sequence of depth-first searches or iter-ations. Before each iteration, a threshold is set and search is discontinued whenthis threshold is reached in each expanded path. The idea of iterative deepeningwas applied to heuristic search by Korf in [9] (algorithm IDA∗). The main ideais that the threshold for iteration i+ 1 is set as the minimum scalar f(n) value

266 J. Coego, L. Mandow, and J.L. Pérez de la Cruz

of the nodes n at which search was discontinued in iteration i. Later this ideawas generalized to multiobjective search and in this way algorithm IDMOA∗ wasdefined by Harikumar and Kumar in [8].

IDMOA∗ focuses on a single objective at a time, so it also maintains a scalarthreshold to test generated nodes in order to discontinue search. This implies thattests against the current threshold will be carried out quickly. Initially IDMOA∗focuses on the first objective, applying iterative deepening until it gets the setof non-dominated solutions that have the smallest value for the first objective.The same scheme applies to the remaining objectives, with the exception thateach of them will also have an upper limit, given by the maximum value of suchobjectives in all solutions found so far. IDMOA∗ is proven to be admissible, thatis, it finds the whole set of non-dominated solutions.

The main drawback of this algorithm is that while setting an objective, itdoes not process or take into account values that appear during the expansionsfor the rest of pending objectives. Besides this, although the dominance testswith the threshold vector remain simple, IDMOA∗ must include extra teststo delete non-dominated solutions possibly added to the solution set in priorsteps.

These drawbacks of IDMOA∗ lead us to present a new approach to multiob-jective iterative deepening. The resulting algorithm is called PIDMOA∗ (ParetoFront Iterative Deepening Multiobjective A∗).Contrary to IDMOA∗, PIDMOA∗takes into account all objectives simultaneously. That means that in each iter-ation we consider a set Threshold of non-dominated vectors and search is dis-countinued at any node n such that its vector valued cost is dominated by avector in Threshold.

The arguments of PIDMOA∗ are a graph G, a start node s and a set of goalnodes Γ . It maintains a set SOLUTION of found solutions (initialized to ∅)and a set of thresholds Threshold (initialized to the subset of non-dominatedheuristic vectors of the start node). Each solution is a pair (γ,f(γ)) where γ isa solution node and f(γ) ∈ F (γ) is the value of a solution path to γ. Succesiveiterations are defined by the actualizations of the set of cost vectors Threshold.When the set is empty, the algorithm terminates.

Actualizations of Threshold are done by performing a depth-first search DFSstarting from s. During this search, when reaching a node n the following tests aredone: (i) if n is fully dominated by previously found solutions, n is discarded andsearch is discontinued at n; (ii) if f(n) is greater than some current threshold, itis accumulated as a threshold for the next iteration and search is discontinuedat n; (iii) if n is a goal node, n is added to SOLUTION; (iv) if none of the aboveconditions hold, search is continued in a depth-first fashion.

3.1 Example

Figure 1 shows a sample tree search problem, where each arc is labelled witha vector cost (2 objectives), and each node is labelled with an heuristic vec-tor estimate. Search starts at the root node. PIDMOA* would set its threshold

A New Approach to Iterative Deepening Multiobjective A* 267

initially to {(5, 5)}, i.e. the heuristic estimate of the root. In its 1st iteration,all nodes with estimates non-dominated by the Threshold would be explored ina depth first fashion. The set of explored nodes and the Threshold of iteration1 are shown in figure 2. At the 2nd iteration, the Threshold would is updatedto {(7, 5), (5, 7)}, i.e. the non-dominated set of all path costs that exceeded theprevious threshold at iteration 1. The new threshold, and the set of explorednodes in the 2nd iteration is shown in figure 3. Subsequent iterations perform ina similar fashion and are shown in figures 4-5. A solution is found at iteration 3(cost (10, 5)). All vectors dominated by this solution are discarded from furtherthresholds. Search is also discontinued at nodes with cost dominated by solutionsalready found. A 2nd solution is found at iteration 4 (cost (5, 10)), and thethreshold becomes empty. Solutions are found at iteration 4, and the Thresholdbecomes empty. Next we can see the pseudocode of the algorithm.

PIDMOA∗ (G, s, Γ )SOLUTION = ∅; Threshold = {h(s)}WHILE Threshold �= ∅

Threshold = DFS (s, Threshold)return (SOLUTION)

DFS (node, threshold_par)nodomvectorsf = {f(node) ∈ F (node) | (�(γ, P ∗(γ)) ∈ SOLUTION | P ∗(γ) � f(node))}IF (nodomvectorsf = ∅) THEN return ∅;domvectorsf = {f(node) ∈ F (node) | (∃c ∈ threshold_par | c ≺ f(node))}IF (domvectorsf �= ∅) THEN return domvectorsf;IF (node ∈ Γ ) THEN

SOLUTION = SOLUTION ∪ {(node, f(node))}ELSE

threshold_dfs = ∅successors = expand_node (node)FOR each n in successors DO

threshold_dfs = threshold_dfs ∪ DFS(n, threshold_par)delete dominated vectors within threshold_dfs;

return (threshold_dfs)

Fig. 1. Multiobjective problem

268 J. Coego, L. Mandow, and J.L. Pérez de la Cruz

Fig. 2. 1st iteration of PIDMOA∗

Fig. 3. 2nd iteration of PIDMOA∗

A New Approach to Iterative Deepening Multiobjective A* 269

Fig. 4. 3rd iteration of PIDMOA∗

Fig. 5. 4th iteration of PIDMOA∗

270 J. Coego, L. Mandow, and J.L. Pérez de la Cruz

3.2 Properties of PIDMOA∗

We will make the following assumptions:1. The graph is connected and the branching factor is bounded.2. There exists at least one solution.3. For each objective i there exists a minimum positive edge cost εi such that

for every edge cost c, 0 < εi ≤ ci∀i.4. The heuristic values along any solution path P ∗ are all non-negative.5. The multiobjective heuristic function H(n) is admissible. A multiobjective

heuristic function H(n) is admissible when for all non-dominated solutionpaths P ∗ = (s,n1, ...,ni,ni+1, ...,γk), γk ∈ Γ and each subpath P ∗i = (s,n1,...,ni) of P ∗, there exists h ∈ H(ni) such that g(P ∗i ) + h � g(P ∗).

It follows from assumptions 2 and 3 that the set of non-dominated solutions isnonempty and finite.Lemma 1. For every iteration i and every threshold ti+1 ∈ Thresholdi+1, thereexists a ti ∈ Thresholdi such that ti ≺ ti+1.Proof: Trivial from PIDMOA∗ code.

Lemma 2. Nodes expanded in iteration i will be also expanded in iteration i+1.

Proof: If n was expanded in interation i, then for all ti ∈ Thresholdi, f (n) ∼ tior f(n) � ti. From Lemma 1, we have that ti ≺ ti+1, hence f(n) ∼ ti+1, orf(n) � ti ≺ ti+1, hence f(n) ≺ ti+1. That is, for all ti+1 ∈ Thresholdi+1,f(n) ∼ ti+1 or f(n) ≺ ti+1, so n will be expanded in iteration i+ 1.

Lemma 3. For every non-dominated solution path P ∗ = (s = n0, n1, . . . , ni,. . . , γ) and every subpath P ∗i = (s = n0, n1, . . . , ni), f(P ∗i ) � f(P ∗).

Proof: f(P ∗i ) = g(P ∗i ) +h(P ∗i ),h(P ∗i ) ∈ H(ni) and f (P ∗) = g(P ∗) +h(P ∗) =g(P ∗) + h(γ). But h(γ) = 0, hence f(P ∗) = g(P ∗) = g(P ∗i ) + g(P ∗r ) whereP ∗r = (ni, ni+1, . . . , γ). Since H(n) is admissible (assumption 5), there existsh(ni) ∈ H(ni) such that g(P ∗i ) + h(ni) � g(P ∗). Substracting g(P ∗i ) fromboth terms, h(ni) � g(P ∗r ). So it exists an h(ni) ∈ H(ni) such that f(P ∗i ) =h(ni) + g(P ∗i ) � g(P ∗i ) + g(P ∗r ) = g(P ∗) = f(P ∗).

Lemma 4. In each iteration i of PIDMOA∗, for all γ ∈ Γ such that (γ,f(γ)) /∈SOLUTION, there exists a t ∈ Thresholdi such that t ≺ f (γ).

Proof: Let us assume that there does not exist t ∈ Thresholdi such that t ≺ f (γ).Let P ∗ = (s, n1, . . . , γ). From Lemma 3, we have that for all P ∗i ⊂ P ∗, f(P ∗i ) �f(P ∗). Since γ was not found in iteration i, for all t ∈ Thresholdi, f(P ∗) ∼ tor f (P ∗) � t. Therefore for all t ∈ Thresholdi, f(P ∗i ) ∼ t or f(P ∗i ) � t. Everyni ∈ P ∗ will be expanded, except γ, which will be added to SOLUTION.

Lemma 5. If there exists P ∗(γj) = (s, . . . , γj), γj ∈ Γ such that (γj ,f(P ∗γj )) ∈SOLUTION , then it does not exist P ∗(γi) = (s, . . . , γi), γi ∈ Γ such thatf(P ∗γi) ≺ f (P ∗γj ). That is, for all (γj ,f(P ∗γj )) ∈ SOLUTION , (γj ,f(P ∗γj )) ∈C∗; all the solutions found by PIDMOA∗ are non-dominated solutions.

A New Approach to Iterative Deepening Multiobjective A* 271

Proof: Let us assume that there exists γi ∈ Γ such that f (P ∗γi) ≺ f(P ∗γj ). If(γi,f(P ∗γi)) ∈ SOLUTION, then (γj ,f(P ∗γj )) will never join SOLUTION becauseof the first test in PIDMOA∗ code (associated to variable nodomvectorsf ). On theother hand, if (γi,f(P ∗γi)) /∈ SOLUTION, since (γj ,f(P ∗γj )) ∈ SOLUTION, then(a) for all t ∈ Thresholdi, P ∗γj ∼ t, or P ∗γj ≺ t. But (γi,f(P ∗γi)) /∈ SOLUTION,therefore there exists t ∈ Thresholdi such that t ≺ f(P ∗γi). So, there existst ∈ Thresholdi such that t ≺ f (P ∗γi) ≺ f(P ∗γj ), which contradicts (a).

Theorem 1. When PIDMOA∗ finishes, SOLUTION = C∗, that is, every nondominated solution will be found by PIDMOA∗.

Proof: Let us assume that there exists (γx,f (P ∗γx)) ∈ C∗ such that (γx,f (P ∗γx))/∈ SOLUTION . From Lemma 4, there exists t ∈ Tresholdn such that t ≺f(P ∗γx). Since (γx,f(P ∗γx)) ∈ C∗, f(P ∗γx) will be returned from DFS call andThresholdn will not be empty and hence PIDMOA∗ will not finish.

Theorem 2. PIDMOA∗ finishes and at its termination SOLUTION = C∗.

Proof: By theorem 1, it suffices to prove that PIDMOA∗ always finishes. Letcmax = (v1, . . . , vk), where vi = max{yi} and yi is the i-th objective value forevery f (P ∗γx) included in SOLUTION. Then for all (γi,f(P ∗γi)) ∈ SOLUTIONit holds that f(P ∗γi) ≺ cmax. By assumptions 1 and 3, each expanded pathwill reach a cost of cmax in at most �max{vi}

max{εi}� steps. At step �max{vi}max{εi}�, each

expanded node will verify that there exists (γ,f(P ∗γ )) ∈ SOLUTION such thatf(P ∗γ ) ≺ f (node) and, as a result of this, the set of thresholds for the nextiteration will be empty, so PIDMOA∗ will finish.

4 Empirical Evaluation

Several rounds of empirical evaluations were carried out to compare the perfor-mance of IDMOA* and PIDMOA*. Complete binary trees of increasing depthwere generated up to depth 20. Arcs were labeled with integer random costs inthe range [1,5] for two different objectives. Approximately 10% of the leaf nodeswere designated as goal nodes in each tree. Both algorithms were run withoutheuristic (blind search), and using the difference between tree depth and nodedepth as an optimistic heuristic estimate for all nodes and both objective func-tions (heuristic search). This experimental setup is easy to reproduce. All resultspresented below are averaged for three different problems for each tree depth.

Figure 6a shows the time requirements of both algorithms. All of them show asharp increase in time requirements, in accordance with the exponential growthrate of the explored graphs. Therefore these results can be better appreciated us-ing a logarithmic scale for the time axis (figure 6b). Three important conclusionscan be drawn. In the first place, the time requirements of BIDMOA are morethan an order of magnitude smaller than those of IDMOA for the hardest prob-lems considered (depth 20). Furthermore, the growth rate of time requirementswith tree depth appears to be larger for IDMOA than for BIDMOA. Finally,

272 J. Coego, L. Mandow, and J.L. Pérez de la Cruz

(a)

(b)

(c)Fig. 6. (a) Time requirements with linear time scale; (b) Time requirements withlogarithmic time scale; (c) Node expansions with logarithmic node scale

A New Approach to Iterative Deepening Multiobjective A* 273

BIDMOA makes better use of heuristic information, achieving a comparativelylarger reduction in time requirements when compared to blind search. An ex-planation of this behavior can be found in figure 6c, which shows the number ofnode expansions performed by the algorithms (logarithmic scale was used for thevertical axis). Analogous behavior of the algorithms was observed in additionaltest sets with larger cost ranges (the largest range tested was [1,100]).

5 Conclusions and Future Work

The paper presents PIDMOA*, a natural extension of the heuristic search algo-rithm IDA* to the multiobjective case. The algorithm is shown to be optimaland complete under reasonable asumptions. PIDMOA* optimizes all objectivessimultaneously, and performs iterative deepening using dominance checks over aset of vector-valued bounds to discontinue search at each deepening stage. Thisis in contrast with IDMOA*, that sequentially optimizes each objective, anduses scalar bounds for each deepening stage. Experimental tests over randomlygenerated binary trees with two objectives show that the deepening strategy ofPIDMOA* clearly outperforms IDMOA* in time requirements and node expan-sions, and makes better use of heuristic information.

Future work includes deeper formal and experimental analyses of the perfor-mance of PIDMOA*, and comparison to other multiobjective search algorithms.

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