{"type":"ScholarlyArticle","title":"On Computational Complexity of Pickup-and-Delivery Problems with Precedence Constraints or Time Windows","authors":[{"name":"Xing Tan","slug":"xing-tan","given":"Xing","family":"Tan","affiliation":"Information Retrieval and Knowledge Management Research Lab , York University , Canada"},{"name":"Jimmy Xiangji Huang","slug":"jimmy-xiangji-huang","given":"Jimmy Xiangji","family":"Huang","affiliation":"Information Retrieval and Knowledge Management Research Lab , York University , Canada"}],"abstract":"Pickup-and-Delivery (PD) problems consider routing vehicles to achieve a set of tasks related to ``Pickup'', and to  ``Delivery''. Meanwhile these tasks might subject to Precedence Constraints (PDPC) or Time Windows (PDTW). PD is a variant to Vehicle Routing Problems (VRP), which have been extensively studied for decades. In the recent years, PD demonstrates its closer relevance to AI. With an awareness that few work has been dedicated so far in addressing where the tractability boundary line can be drawn for PD problems, we identify in this paper a set of highly restricted PD problems and prove their NP-completeness. Many problems from a multitude of applications and industry domains are general versions of PDPC. Thus this new result of NP-hardness, of PDPC, not only clarifies the computational complexity of these problems, but also sets up a firm base for the requirement on use of approximation or heuristics, as opposed to looking for exact but intractable algorithms for solving them. We move on to perform an empirical study to locate sources of intractability in PD problems. That is, we propose a local-search formalism and algorithm for solving PDPC problems in particular. Experimental results  support  strongly effectiveness and efficiency of the local-search. Using the local-search as a solver for randomly generated PDPC problem instances, we obtained interesting and potentially useful insights regarding computational hardness of PDPC and PD.","doi":"10.24963/ijcai.2019/782","url":"https://ijcai.org/proceedings/2019/782","pdf":"https://ijcai.org/proceedings/2019/0782.pdf","markdown":"https://ijcai.org/proceedings/2019/782.md","bibtex":"https://ijcai.org/proceedings/2019/bibtex/782","volume":{"slug":"2019","title":"Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2019","url":"https://ijcai.org/proceedings/2019/","year":2019,"location":"Macao","dates":"10-16 August 2019","editors":"Sarit Kraus","isbn":"978-0-9992411-4-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2019"},"section":"Planning and Scheduling","track":"Main track","number":782,"pages":{"first":5635,"last":5643},"keywords":["Planning and Scheduling: Scheduling","Agent-based and Multi-agent Systems: Resource Allocation","Knowledge Representation and Reasoning: Computational Complexity of Reasoning","Robotics: Motion and Path Planning","Planning and Scheduling: Planning and Scheduling"],"videos":[],"references":[{"position":1,"raw":"[Allen, 1983] James Allen. Maintaining knowledge about temporal intervals. Communications of the ACM, 26(11):832-843, 1983.","title":"Maintaining knowledge about temporal intervals","authors":"James Allen","year":"1983","venue":"Communications of the ACM","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Baldacci et al., 2011] Roberto Baldacci, Enrico Bartolini, and Aristide Mingozzi. An exact algorithm for the pickup and delivery problem with time windows. Operations Re- search, 59(2):414-426, 2011.","title":"An exact algorithm for the pickup and delivery problem with time windows","authors":"Roberto Baldacci, Enrico Bartolini, Aristide Mingozzi","year":"2011","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Battarra et al., 2014] Maria Battarra, Jean-Franc ¸ois Cordeau, and Manuel Iori. Chapter 6: Pickup-and- delivery problems for goods transportation. In Vehicle Routing: Problems, Methods, and Applications, Second Edition, pages 161-191. SIAM, 2014.","title":"Chapter 6: Pickup-anddelivery problems for goods transportation","authors":"Maria Battarra, Jean-Franc ¸ois Cordeau, Manuel Iori","year":"2014","venue":"Vehicle Routing: Problems, Methods, and Applications, Second Edition","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Beck et al., 2003] J Christopher Beck, Patrick Prosser, and Evgeny Selensky. Vehicle routing and job shop schedul- ing: What's the difference? In ICAPS, pages 267-276, 2003.","title":"Vehicle routing and job shop scheduling: What's the difference?","authors":"Beck","year":"2003","venue":"ICAPS","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bent and Van Hentenryck, 2007a] Russell Bent and Pascal Van Hentenryck. Randomized adaptive spatial decoupling for large-scale vehicle routing with time windows. In AAAI, volume 7, pages 173-178, 2007.","title":"Randomized adaptive spatial decoupling for large-scale vehicle routing with time windows","authors":"Van Bent, Russell Hentenryck, Pascal Bent, Van Hentenryck","year":"2007","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Bent and Van Hentenryck, 2007b] Russell Bent and Pascal Van Hentenryck. Waiting and relocation strategies in on- line stochastic vehicle routing. In IJCAI, pages 1816- 1821, 2007.","title":"Waiting and relocation strategies in online stochastic vehicle routing","authors":"Van Bent, Russell Hentenryck, Pascal Bent, Van Hentenryck","year":"2007","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2007/293"},{"position":7,"raw":"[Bistaffa et al., 2015] Filippo Bistaffa, Alessandro Farinelli, and Sarvapali D. Ramchurn. Sharing rides with friends: A coalition formation algorithm for ridesharing. In AAAI, pages 608-614, 2015.","title":"Sharing rides with friends: A coalition formation algorithm for ridesharing","authors":"Filippo Bistaffa, Alessandro Farinelli, Sarvapali D Ramchurn","year":"2015","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Blum and Rivest, 1988] Avrim Blum and Ronald L Rivest. Training a 3-node neural network is NP-complete. In NIPS, pages 494-501, 1988.","title":"Training a 3-node neural network is NP-complete","authors":"Avrim Blum, Ronald L Rivest","year":"1988","venue":"NIPS","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Bräysy and Gendreau, 2005] Olli Bräysy and Michel Gen- dreau. Vehicle routing problem with time windows, Part II: Metaheuristics. Transportation Science, 39(1):119-139, 2005.","title":"Olli Bräysy and Michel Gendreau. Vehicle routing problem with time windows, Part II: Metaheuristics","authors":"Gendreau Bräysy","year":"2005","venue":"Transportation Science","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Bylander, 1994] Tom Bylander. The computational com- plexity of propositional STRIPS planning. Artificial In- telligence, 69(1-2):165-204, 1994.","title":"The computational complexity of propositional STRIPS planning","authors":"Tom Bylander, Bylander","year":"1994","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Cohen and Beck, 2017a] Eldan Cohen and J. Christopher Beck. Cost-based heuristics and node re-expansions across the phase transition. In SOCS, pages 11-19, 2017.","title":"Cost-based heuristics and node re-expansions across the phase transition","authors":"Beck ; A] Eldan Cohen, J Christopher Cohen, Beck","year":"2017","venue":"SOCS","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Cohen and Beck, 2017b] Eldan Cohen and J. Christopher Beck. Problem difficulty and the phase transition in heuris- tic search. In AAAI, pages 780-786, 2017.","title":"Problem difficulty and the phase transition in heuristic search","authors":"Beck ; B] Eldan Cohen, J Christopher Cohen, Beck","year":"2017","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Coltin and Veloso, 2014] Brian Coltin and Manuela Veloso. Scheduling for transfers in pickup and delivery problems with very large neighborhood search. In AAAI, pages 2250-2256, 2014.","title":"Scheduling for transfers in pickup and delivery problems with very large neighborhood search","authors":"Brian Coltin, Manuela Veloso","year":"2014","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Cooper, 1990] Gregory F Cooper. The computational com- plexity of probabilistic inference using Bayesian belief networks. Artificial Intelligence, 42(2-3):393-405, 1990.","title":"The computational complexity of probabilistic inference using Bayesian belief networks","authors":"F Gregory, Cooper","year":"1990","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Cordeau, 2006] Jean-Franc ¸ois Cordeau. A branch-and-cut algorithm for the dial-a-ride problem. Operations Re- search, 54(3):573-586, 2006.","title":"A branch-and-cut algorithm for the dial-a-ride problem","authors":"Jean-Franc ¸ois Cordeau","year":"2006","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Dean and Boddy, 1988] Thomas Dean and Mark Boddy. Reasoning about partially ordered events. Artificial Intel- ligence, 36(3):375-399, 1988.","title":"Reasoning about partially ordered events","authors":"Thomas Dean, Mark Boddy","year":"1988","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Doerner and Salazar-González, 2014] Karl F Doerner and Juan-José Salazar-González. Pickup-and-delivery prob- lems for people transportation. In Vehicle Routing: Prob- lems, Methods, and Applications, Second Edition, pages 193-212. SIAM, 2014.","title":"Pickup-and-delivery problems for people transportation","authors":"F Karl, Juan-José Doerner, Salazar-González","year":"2014","venue":"Vehicle Routing: Problems, Methods, and Applications, Second Edition","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Dumas et al., 1991] Yvan Dumas, Jacques Desrosiers, and Franc ¸ois Soumis. The pickup and delivery problem with time windows. European Journal of Operational Re- search, 54(1):7 -22, 1991.","title":"The pickup and delivery problem with time windows","authors":"Yvan Dumas, Jacques Desrosiers, Franc ¸ois Soumis","year":"1991","venue":"European Journal of Operational Research","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Garey and Johnson, 1979] Michael Garey and David John- son. Computers and intractability -a guide to NP- completeness. W.H. Freeman and Company, 1979.","title":"Computers and intractability -a guide to NPcompleteness","authors":"Michael Garey, David Johnson","year":"1979","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Gendreau et al., 1994] Michel Gendreau, Alain Hertz, and Gilbert Laporte. A tabu search heuristic for the vehi- cle routing problem. Management Science, 40(10):1276- 1290, 1994.","title":"A tabu search heuristic for the vehicle routing problem","authors":"Michel Gendreau, Alain Hertz, Gilbert Laporte","year":"1994","venue":"Management Science","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Gini, 2017] Maria L Gini. Multi-robot allocation of tasks with temporal and ordering constraints. In AAAI, pages 4863-4869, 2017.","title":"Multi-robot allocation of tasks with temporal and ordering constraints","authors":"Maria L Gini","year":"2017","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Gomes et al., 2008] Carla P Gomes, Henry Kautz, Ashish Sabharwal, and Bart Selman. Satisfiability solvers. Foun- dations of Artificial Intelligence, 3:89-134, 2008.","title":"Satisfiability solvers","authors":"Carla P Gomes, Henry Kautz, Ashish Sabharwal, Bart Selman","year":"2008","venue":"Foundations of Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Hernández-Pérez and Salazar-González, 2009] Hipólito Hernández-Pérez and Juan-José Salazar-González. The multi-commodity one-to-one pickup-and-delivery travel- ing salesman problem. European Journal of Operational Research, 196(3):987-995, 2009.","title":"The multi-commodity one-to-one pickup-and-delivery traveling salesman problem","authors":"Salazar-González ; Hernández-Pérez, Juan-José Hernández-Pérez, Salazar-González","year":"2009","venue":"European Journal of Operational Research","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Johnson et al., 1988] David S. Johnson, Christos H. Pa- padimitriou, and Mihalis Yannakakis. How easy is lo- cal search? Journal of Computer and System Sciences, 37(1):79 -100, 1988.","title":"How easy is local search","authors":"David S Johnson, Christos H Papadimitriou, Mihalis Yannakakis","year":"1988","venue":"Journal of Computer and System Sciences","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Krokhin et al., 2003] Andrei Krokhin, Peter Jeavons, and Peter Jonsson. Reasoning about temporal relations: The tractable subalgebras of Allen's interval algebra. J. ACM, 50(5):591-640, September 2003.","title":"Reasoning about temporal relations: The tractable subalgebras of Allen's interval algebra","authors":"Krokhin","year":"2003","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Laporte et al., 2006] Gilbert Laporte, Michel Gendreau, Jean-Yves Potvin, and Frédéric Semet. Classical and mod- ern heuristics for the vehicle routing problem. Interna- tional Transactions in Operational Research, 7(4-5):285- 300, 2006.","title":"Classical and modern heuristics for the vehicle routing problem","authors":"Gilbert Laporte, Michel Gendreau, Jean-Yves Potvin, Frédéric Semet","year":"2006","venue":"International Transactions in Operational Research","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Maciejewski et al., 2017] Michal Maciejewski, Joschka Bischoff, Sebastian Hörl, and Kai Nagel. Towards a testbed for dynamic vehicle routing algorithms. In AAMAS, pages 69-79, 2017.","title":"Towards a testbed for dynamic vehicle routing algorithms","authors":"Michal Maciejewski, Joschka Bischoff, Sebastian Hörl, Kai Nagel","year":"2017","venue":"AAMAS","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Minton et al., 1992] Steven Minton, Mark D. Johnston, An- drew B. Philips, and Philip Laird. Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems. Artificial Intelligence, 58(1):161 - 205, 1992.","title":"Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems","authors":"Steven Minton, Mark D Johnston, Andrew B Philips, Philip Laird","year":"1992","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Möhring, 1989] Rolf H Möhring. Computationally tractable classes of ordered sets. In Algorithms and Order, pages 105-193. Springer, 1989.","title":"Computationally tractable classes of ordered sets","authors":"Rolf H Möhring","year":"1989","venue":"Algorithms and Order","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Nebel and Bürckert, 1995] Bernhard Nebel and Hans- Jürgen Bürckert. Reasoning about temporal relations: A maximal tractable subclass of Allen's interval algebra. J. ACM, 42(1):43-66, 1995.","title":"Reasoning about temporal relations: A maximal tractable subclass of Allen's interval algebra","authors":"Bürckert Nebel, Bernhard Nebel, Hans-Jürgen Bürckert","year":"1995","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Nunes et al., 2017] Ernesto Nunes, Marie D. Manner, Hakim Mitiche, and Maria L. Gini. A taxonomy for task allocation problems with temporal and ordering con- straints. Robotics and Autonomous Systems, 90:55-70, 2017.","title":"A taxonomy for task allocation problems with temporal and ordering constraints","authors":"Ernesto Nunes, Marie D Manner, Hakim Mitiche, Maria L Gini","year":"2017","venue":"Robotics and Autonomous Systems","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Parragh et al., 2008] Sophie N Parragh, Karl F Doerner, and Richard F Hartl. A survey on pickup and delivery prob- lems. Journal für Betriebswirtschaft, 58(1):21-51, 2008.","title":"A survey on pickup and delivery problems","authors":"Karl F Sophie N Parragh, Richard F Doerner, Hartl","year":"2008","venue":"Journal für Betriebswirtschaft","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Pham et al., 2008] Duc Nghia Pham, John Thornton, and Abdul Sattar. Modelling and solving temporal reason- ing as propositional satisfiability. Artificial Intelligence, 172(15):1752 -1782, 2008.","title":"Modelling and solving temporal reason-ing as propositional satisfiability","authors":"Nghia Duc, John Pham, Abdul Thornton, Sattar","year":"2008","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Potvin and Bengio, 1996] Jean-Yves Potvin and Samy Ben- gio. The vehicle routing problem with time windows part II: Genetic search. INFORMS Journal on Computing, 8(2):165-172, 1996.","title":"The vehicle routing problem with time windows part II: Genetic search","authors":"Bengio ; Jean-Yves Potvin, Samy Potvin, Bengio","year":"1996","venue":"INFORMS Journal on Computing","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Savelsbergh and Sol, 1995] Martin Savelsbergh and Marc Sol. The general pickup and delivery problem. Trans- portation Science, 29(1):17-29, 1995.","title":"Martin Savelsbergh and Marc Sol. The general pickup and delivery problem","authors":"Sol Savelsbergh","year":"1995","venue":"Transportation Science","doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Savelsbergh, 1985] M. W. P. Savelsbergh. Local search in routing problems with time windows. Annals of Opera- tions Research, 4(1):285-305, Dec 1985.","title":"Local search in routing problems with time windows","authors":"M W P Savelsbergh","year":"1985","venue":"Annals of Operations Research","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Shang and Wah, 1998] Yi Shang and Benjamin W Wah. A discrete Lagrangian-based global-search method for solv- ing satisfiability problems. Journal of global optimization, 12(1):61-99, 1998.","title":"A discrete Lagrangian-based global-search method for solving satisfiability problems","authors":"Yi Shang, Benjamin W Wah","year":"1998","venue":"Journal of global optimization","doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Shaw, 1998] Paul Shaw. Using constraint programming and local search methods to solve vehicle routing problems. In CP, pages 417-431, 1998.","title":"Using constraint programming and local search methods to solve vehicle routing problems","authors":"Paul Shaw","year":"1998","venue":"CP","doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Taillard et al., 1997] Éric Taillard, Philippe Badeau, Michel Gendreau, Franc ¸ois Guertin, and Jean-Yves Potvin. A tabu search heuristic for the vehicle routing problem with soft time windows. Transportation Science, 31(2):170-186, 1997.","title":"A tabu search heuristic for the vehicle routing problem with soft time windows","authors":"Éric Taillard, Philippe Badeau, Michel Gendreau, Jean-Yves Franc ¸ois Guertin, Potvin","year":"1997","venue":"Transportation Science","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Tan and Gruninger, 2009] Xing Tan and Michael Gruninger. Towards tractable reasoning on temporal projection problems. In Proceedigns of the IEEE In- ternational Conference on Intelligent Computing and Intelligent Systems, volume 1, pages 723-727, 2009.","title":"Towards tractable reasoning on temporal projection problems","authors":"Xing Tan, Michael Gruninger","year":"2009","venue":"Proceedigns of the IEEE International Conference on Intelligent Computing and Intelligent Systems","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"[Tan and Gruninger, 2014] Xing Tan and Michael Gruninger. The complexity of partial-order plan via- bility problems. In ICAPS, pages 307-313, 2014.","title":"The complexity of partial-order plan viability problems","authors":"Xing Tan, Michael Gruninger","year":"2014","venue":"ICAPS","doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Tan, 2012] Xing Tan. The Application of Ontologies to Rea- soning with Process Modeling Formalisms. PhD thesis, University of Toronto, 2012.","title":"The Application of Ontologies to Reasoning with Process Modeling Formalisms","authors":"Xing Tan","year":"2012","venue":null,"doi":null,"url":null,"internal_url":null},{"position":43,"raw":"[Toth and Vigo, 2014] Paolo Toth and Daniele Vigo. Vehicle Routing: Problems, Methods, and Applications. SIAM, 2014.","title":"Vehicle Routing: Problems, Methods, and Applications","authors":"Paolo Toth, Daniele Vigo","year":"2014","venue":null,"doi":null,"url":null,"internal_url":null},{"position":44,"raw":"[Vaessens et al., 1996] R. J. M. Vaessens, E. H. L. Aarts, and J. K. Lenstra. Job shop scheduling by local search. IN- FORMS Journal on Computing, 8(3):302-317, 1996.","title":"Job shop scheduling by local search","authors":"R J M Vaessens, E H L Aarts, J K Lenstra","year":"1996","venue":"IN-FORMS Journal on Computing","doi":null,"url":null,"internal_url":null},{"position":45,"raw":"[Vilain and Kautz, 1986] Marc B. Vilain and Henry A. Kautz. Constraint propagation algorithms for temporal reasoning. In AAAI, pages 377-382, 1986.","title":"Constraint propagation algorithms for temporal reasoning","authors":"Marc B Vilain, Henry A Kautz","year":"1986","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":46,"raw":"[Vokřínek et al., 2010] Jiří Vokřínek, Antonín Komenda, and Michal Pěchouček. Agents towards vehicle routing prob- lems. In AAMAS, pages 773-780, 2010.","title":"Agents towards vehicle routing problems","authors":"Vokřínek","year":"2010","venue":"AAMAS","doi":null,"url":null,"internal_url":null},{"position":47,"raw":"[Yao et al., 2018] Huaxiu Yao, Fei Wu, Jintao Ke, Xianfeng Tang, Yitian Jia, Siyu Lu, Pinghua Gong, Jieping Ye, and Zhenhui Li. Deep multi-view spatial-temporal network for taxi demand prediction. In AAAI, pages 2588-2595, 2018.","title":"Deep multi-view spatial-temporal network for taxi demand prediction","authors":"Huaxiu Yao, Fei Wu, Jintao Ke, Xianfeng Tang, Yitian Jia, Siyu Lu, Pinghua Gong, Jieping Ye, Zhenhui Li","year":"2018","venue":"AAAI","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}