{"type":"ScholarlyArticle","title":"Improved Algorithms for Allen's Interval Algebra by Dynamic Programming with Sublinear Partitioning","authors":[{"name":"Leif Eriksson","slug":"leif-eriksson","given":"Leif","family":"Eriksson","affiliation":"Department of Computer and Information Science , Linköping University , Linköping , Sweden"},{"name":"Victor Lagerkvist","slug":"victor-lagerkvist","given":"Victor","family":"Lagerkvist","affiliation":"Department of Computer and Information Science , Linköping University , Linköping , Sweden"}],"abstract":"Allen's interval algebra is one of the most well-known calculi in qualitative temporal reasoning with numerous applications in artificial intelligence. Very recently, there has been a surge of improvements in the fine-grained complexity of NP-hard reasoning tasks in this algebra, which has improved the running time from the naive 2^O(n^2) to O*((1.0615n)^n), and even faster algorithms are known for unit intervals and the case when we a bounded number of overlapping intervals. \r\nDespite these improvements the best known lower bound is still only 2^o(n) under the exponential-time hypothesis and major improvements in either direction seemingly require fundamental advances in computational complexity. \r\nIn this paper we propose a novel framework for solving NP-hard qualitative reasoning problems which we refer to as dynamic programming with sublinear partitioning. \r\n Using this technique we obtain a major improvement of O*((cn/log(n))^n) for Allen's interval algebra. To demonstrate that the technique is applicable to further problem domains we apply it to a problem in qualitative spatial reasoning, the cardinal direction calculus, and solve it in O*((cn/log(n))^(2n/3)) time. Hence, not only do we significantly advance the state-of-the-art for NP-hard qualitative reasoning problems, but obtain a novel algorithmic technique that is likely applicable to many problems where 2^O(n) time algorithms are unlikely.","doi":"10.24963/ijcai.2023/213","url":"https://ijcai.org/proceedings/2023/213","pdf":"https://ijcai.org/proceedings/2023/0213.pdf","markdown":"https://ijcai.org/proceedings/2023/213.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/213","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Constraint Satisfaction and Optimization","track":"Main Track","number":213,"pages":{"first":1919,"last":1926},"keywords":["Constraint Satisfaction and Optimization: CSO: Constraint satisfaction","Knowledge Representation and Reasoning: KRR: Qualitative, geometric, spatial, and temporal reasoning"],"videos":[],"references":[{"position":1,"raw":"[Allen and Koomen, 1983] James F. Allen and Johannes A. G. M. Koomen. Planning using a temporal world model. In Proceedings of the 8th International Joint Conference on Artificial Intelligence (IJCAI-1983), pages 741-747.","title":"Planning using a temporal world model","authors":"James F Allen, Johannes A G M Koomen","year":null,"venue":"Proceedings of the 8th International Joint Conference on Artificial Intelligence (IJCAI-1983)","doi":null,"url":null,"internal_url":"/proceedings/1983/180"},{"position":2,"raw":"William Kaufmann, 1983.","title":null,"authors":"William Kaufmann","year":"1983","venue":null,"doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Dabrowski et al., 2020] Konrad K. Dabrowski, Peter Jons- son, Sebastian Ordyniak, and George Osipov. Fine-grained complexity of temporal problems. In Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR-2020), pages 284-293, 2020.","title":"Fine-grained complexity of temporal problems","authors":"Konrad K Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov","year":"2020","venue":"Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR-2020)","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Denis and Muller, 2011] Pascal Denis and Philippe Muller. Predicting globally-coherent temporal structures from texts via endpoint inference and graph decomposition. In Pro- ceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI-2011), pages 1788-1793. IJ- CAI/AAAI, 2011.","title":"Predicting globally-coherent temporal structures from texts via endpoint inference and graph decomposition","authors":"Denis, Muller, Pascal Denis, Philippe Muller","year":"2011","venue":"Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI-2011)","doi":null,"url":null,"internal_url":"/proceedings/2011/300"},{"position":5,"raw":"[Dorn, 1995] Jürgen Dorn. Dependable reactive event- oriented planning. Data & Knowledge Engineering, 16(1):27-49, 1995.","title":"Dependable reactive eventoriented planning","authors":"Jürgen Dorn","year":"1995","venue":"Data & Knowledge Engineering","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Dylla et al., 2017] Frank Dylla, Jae H. Lee, Till Mossakowski, Thomas Schneider, André V. Delden, Jasper V. D. Ven, and Diedrich Wolter. A survey of qualitative spatial and temporal calculi: Algebraic and computational properties. ACM Computing Surveys (CSUR), 50(1):7:1-7:39, April 2017.","title":"A survey of qualitative spatial and temporal calculi: Algebraic and computational properties","authors":"Frank Dylla, Jae H Lee, Till Mossakowski, Thomas Schneider, André V Delden, V D Jasper, Diedrich Ven, Wolter","year":"2017","venue":"ACM Computing Surveys (CSUR)","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Eriksson and Lagerkvist, 2021] Leif Eriksson and Victor Lagerkvist. Improved algorithms for Allen's interval alge- bra: a dynamic programming approach. In Proceedings of the 30th International Joint Conference on Artificial Intelligence (IJCAI-2021), pages 1873-1879, 2021.","title":"Improved algorithms for Allen's interval algebra: a dynamic programming approach","authors":"Lagerkvist Eriksson, Leif Eriksson, Victor Lagerkvist","year":"2021","venue":"Proceedings of the 30th International Joint Conference on Artificial Intelligence (IJCAI-2021)","doi":null,"url":null,"internal_url":"/proceedings/2021/258"},{"position":8,"raw":"[Eriksson and Lagerkvist, 2022] Leif Eriksson and Victor Lagerkvist. A multivariate complexity analysis of qualita- tive reasoning problems. In Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI-22, pages 1804-1810. International Joint Confer- ences on Artificial Intelligence Organization, 7 2022.","title":"A multivariate complexity analysis of qualitative reasoning problems","authors":"Lagerkvist Eriksson, Leif Eriksson, Victor Lagerkvist","year":"2022","venue":"Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI-22","doi":null,"url":null,"internal_url":"/proceedings/2022/251"},{"position":9,"raw":"[Frank, 1991] Andrew U. Frank. Qualitative spatial reasoning with cardinal directions. In Hermann Kaindl, editor, Pro- ceedings of the 7th Austrian Conference on Artificial Intelli- gence ( ÖGAI-91), volume 287 of Informatik-Fachberichte, pages 157-167. Springer, 1991.","title":"Qualitative spatial reasoning with cardinal directions","authors":"Andrew U Frank","year":"1991","venue":"Proceedings of the 7th Austrian Conference on Artificial Intelligence ( ÖGAI-91)","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Golumbic and Shamir, 1993] M. C. Golumbic and R. Shamir. Complexity and algorithms for reason- ing about time: A graph-theoretic approach. Journal of the ACM, 40(5):1108-1133, 1993.","title":"Complexity and algorithms for reasoning about time: A graph-theoretic approach","authors":"M C Golumbic, R Shamir","year":"1993","venue":"Journal of the ACM","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Jonsson and Lagerkvist, 2017] Peter Jonsson and Victor Lagerkvist. An initial study of time complexity in infinite- domain constraint satisfaction. Artificial Intelligence, 245:115-133, 2017.","title":"An initial study of time complexity in infinitedomain constraint satisfaction","authors":"Jonsson, Lagerkvist, Peter Jonsson, Victor Lagerkvist","year":"2017","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Jonsson et al., 2021] Peter Jonsson, Victor Lagerkvist, and George Osipov. Acyclic orders, partition schemes and csps: Unified hardness proofs and improved algorithms. Artificial Intelligence, 296:103505, 2021.","title":"Acyclic orders, partition schemes and csps: Unified hardness proofs and improved algorithms","authors":"Jonsson","year":"2021","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Mudrová and Hawes, 2015] Lenka Mudrová and Nick Hawes. Task scheduling for mobile robots using interval algebra. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA-2015), pages 383-388. IEEE, 2015.","title":"Task scheduling for mobile robots using interval algebra","authors":"Hawes Mudrová, Lenka Mudrová, Nick Hawes","year":"2015","venue":"Proceedings of the IEEE International Conference on Robotics and Automation (ICRA-2015)","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Pelavin and Allen, 1987] Richard N. Pelavin and James F. Allen. A model for concurrent actions having temporal extent. In Proceedings of the 6th National Conference on Artificial Intelligence (AAAI-1987), pages 246-250.","title":"A model for concurrent actions having temporal extent","authors":"Richard N Pelavin, James F Allen","year":null,"venue":"Proceedings of the 6th National Conference on Artificial Intelligence (AAAI-1987)","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"Mor- gan Kaufmann, 1987.","title":null,"authors":"Morgan Kaufmann","year":"1987","venue":null,"doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Song and Cohen, 1988] Fei Song and Robin Cohen. The interpretation of temporal relations in narrative. In Pro- ceedings of the 7th National Conference on Artificial Intel- ligence (AAAI-1988), pages 745-750. AAAI Press / The MIT Press, 1988.","title":"Fei Song and Robin Cohen. The interpretation of temporal relations in narrative","authors":"Cohen Song","year":"1988","venue":"Proceedings of the 7th National Conference on Artificial Intelligence (AAAI-1988)","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}