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Computational Complexity of Some Optimization Problems in Planning Linköping Studies in Science and Technology Dissertation No. 1854 Computational Complexity of some Optimization Problems in Planning Meysam Aghighi Linköping University Department of Computer and Information Science SE-581 85 Linköping, Sweden Linköping 2017 c Meysam Aghighi, 2017 ISBN 978-91-7685-519-5 ISSN 0345-7524 URL http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-136280/ Published articles have been reprinted with permission from the respective copyright holder. Typeset using LATEX Printed by LiU-Tryck, Linköping 2017 ii Computational Complexity of some Optimization Problems in Planning ABSTRACT Automated planning is known to be computationally hard in the general case. Propositional planning is PSPACE-complete and first-order planning is undecidable. One method for ana- lyzing the computational complexity of planning is to study restricted subsets of planning in- stances, with the aim of differentiating instances with varying complexity. We use this method- ology for studying the computational complexity of planning. Finding new tractable (i.e. polynomial-time solvable) problems has been a particularly impor- tant goal for researchers in the area. The reason behind this is not only to differentiate between easy and hard planning instances, but also to use polynomial-time solvable instances in order to construct better heuristic functions and improve planners. We identify a new class of tractable cost-optimal planning instances by restricting the causal graph. We also study the computational complexity of oversubscription planning (such as the net-benefit problem) under various restrictions and reveal strong connections with classical planning. Inspired by this, we present a method for compiling oversubscription planning prob- lems into the ordinary plan existence problem. We further study the parameterized complexity of cost-optimal and net-benefit planning under the same restrictions and show that the choice of numeric domain for the action costs has a great impact on the parameterized complexity. We finally consider the parameterized complexity of certain problems related to partial-order planning. In some applications, less restricted plans than total-order plans are needed. There- fore, a partial-order plan is being used instead. When dealing with partial-order plans, one important question is how to achieve optimal partial order plans, i.e. having the highest de- gree of freedom according to some notion of flexibility. We study several optimization problems for partial-order plans, such as finding a minimum deordering or reordering, and finding the minimum parallel execution length. The research presented in this thesis has been partially funded by the National Graduate School of Com- puter Science in Sweden (CUGS). iii Populärvetenskaplig sammanfattning Huvudtemat i denna avhandling är studiet av beräkningskomplexitet för automatisk planering. På en hög nivå kan man betrakta planeringsproblemet så här; man utgår från en värld eller ett system, som ett obemannat fordon eller en autonom robot, och antar att det finns vissa handlingar som ändrar tillståndet i systemet. Man antar dessutom att det finns ett initial- och ett måltillstånd. Planering är problemet att hitta en sekvens av handlingar som tranformerar initialtillståndet till måltillståndet. En sådan sekvens av handlingar kallas en plan. En planerare är ett datorprogram som löser planeringsproblem, och planerare baseras ofta på sökning genom tillstånden i systemet. Vanligtvis vill man optimera sökningen och hitta en plan som är optimal med avseende på något kriterium. Vad man vill optimera beror på tillämpningen men det är ofta att hitta en kortaste eller billigaste plan. En viktig fråga när man studerar automatisk planering är hur snabbt man kan hitta lösningar. Ett sätt att analysera denna fråga är att studera beräkningskomplexiteten för olika planeringsproblem. Komplexitetsteori är den gren av datavetenskapen som försöker klassificera hur svåra olika beräkningsproblem är, det vill säga analysera hur mycket resurser (såsom tid och minne) som behövs. Man likställer ofta effektivt lösbara problem med de som kan lösas i polynomisk tid: ett problem kan lösas i polynomisk tid om det finns en algoritm vars tidsåtgång begränsas av ett polynom i indatas storlek. Det grundläggande planeringsproblemet är exempelvis känt att vara PSPACE-fullständigt. PSPACE är den komplexitetsklass som innehåller alla de beräkningsproblem som kan lösas genom att använda polynomiskt mycket minne och de PSPACE-fullständiga problemen utgör de svåraste problemen i PSPACE. Man tror att det inte finns några PSPACE- fullständiga problem som kan lösas i polynomisk tid—detta är ett viktigt olöst problem inom teoretisk datalogi—och man förväntar sig därför inte att det generella planeringsproblemet kan lösas i polynomisk tid. En metod för att analysera beräkningskomplexitet av planering är att studera begränsade delmängder av planeringsinstanser i syfte att särskilja fall med varierande komplexitet. Vi använder denna metod genomgående i denna avhandling. Så kallad “oversubscription planning” är en typ av planering där man mäter hur nära ett givet måltillstånd v man kan komma. Vi studerar beräkningskomplexiteten för denna typ av planering under olika restriktioner och avslöjar starka kopplingar med vanlig planering. Kostnadsoptimal planering är planering där varje handling har en kostnad och målet är att hitta en plan med minimal kostnad. Dessa handlingskostnader kan hämtas från olika numeriska domäner såsom de positiva heltalen eller de rationella talen. Vi studerar effekter av valet av numeriska domäner och noterar intressanta komplexitetsmässiga skillnader. För kostnadsoptimal planering identifierar vi dessutom en ny klass av instanser som kan lösas i polynomisk tid. Slutligen studerar vi komplexiteten för olika problem inom så kallad partialordningsplanering där man tillåter planer som inte nödvändigtvis är sekvenser av handlingar. vi Acknowledgments First and foremost, I want to thank my advisors Peter Jonsson and Christer Bäckström. It has been an honor to work with and learn from them for the past three years. They have taught me how well the academic research is done. I am specially grateful to Peter for his great help and Christer for his valuable comments in preparing this thesis. The members of TCSLAB and IDA contributed immensely to my per- sonal and professional time as a PhD student at LiU. I specifically thank Si- mon Ståhlberg, who was both a colleague and a friend, Amir Aminifar, Bi- man Roy, Arian Maghazeh, Adrian Horga, Rouhollah Mahfouzi and Victor Lagerkvist. I am also grateful to the administration at IDA that facilitated my work in less direct ways. I specially thank Karin Baardsen, Anne Moe, Marie Jo- hansson, Inger Norén and the director of graduate studies, Professor Petru Eles. Lastly, I would like to thank my family for all their love and encourage- ment. For my parents who raised me with a love of science and supported me in all my pursuits. For my loving, supporting, encouraging and patient wife Zeinab whose support during my whole PhD is so appreciated. Thank you. MEYSAM AGHIGHI Linköping University May 2017 vii List of Papers The thesis is based on the following papers: 1. Meysam Aghighi and Peter Jonsson. Oversubscription planning: Complexity and compilability. In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelli- gence, Québec City, Québec, Canada, July 27 -31, pages 2221-2227, 2014. 2. Meysam Aghighi, Peter Jonsson and Simon Ståhlberg. Tractable cost-optimal planning over restricted polytree causal graphs. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelli- gence, Austin, Texas, USA, January 25-30, pages 3225-3231, 2015. 3. Meysam Aghighi and Christer Bäckström. Cost-optimal and net-benefit planning - A parameterised complexity view. In Proceedings of the Twenty-Fourth International Joint Conference on Arti- ficial Intelligence (IJCAI), Buenos Aires, Argentina, July 25-31, pages 1487- 1493, 2015. 4. Meysam Aghighi and Christer Bäckström. A multi-parameter complexity analysis of cost-optimal and net-benefit planning. In Proceedings of the Twenty-Sixth International Conference on Automated Planning and Scheduling (ICAPS), London, UK, June 12-17, pages 2-10, 2016. 5. Meysam Aghighi and Christer Bäckström. Plan reordering and parallel execution – a parameterized complexity view. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, San Francisco, California, USA, February 4-9, pages 3540-3546, 2017. In addition to the above papers, the author has also contributed to the fol- lowing publications: 1. Meysam Aghighi, Christer Bäckström, Peter Jonsson and Simon Ståhlberg. Analysing Approximability and Heuristics in Planning Using the ix Exponential-time Hypothesis. In Proceedings of the Twenty-second European Conference on Artificial Intel- ligence (ECAI) , the Hague, Holland, August 29th - September 2nd, pages 184-192, 2016. 2. Meysam Aghighi, Christer Bäckström, Peter Jonsson and Simon Ståhlberg. Refining Complexity Analyses in Planning by Exploiting the Exponential-time Hypothesis. In Annals of Mathematics and Artificial Intelligence (AMAI), pages 157-175, 2016. x Contents Abstract iii Populärvetenskaplig Sammanfattning v Acknowledgments vii List of Papers ix Contents xi I Introduction 1 1 Thesis Overview 3 1 About this Thesis . 3 2 Brief Summary of Papers . 4 2 Computational Complexity Theory 7 1 Computational Complexity . 7 2 Parameterized Complexity . 11 3 Automated Planning 15 1 What is Planning? . 15 2 Planning
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