{"type":"ScholarlyArticle","title":"A Multivariate Complexity Analysis of Qualitative Reasoning Problems","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":"Qualitative reasoning is an important subfield of artificial intelligence where one describes relationships with qualitative, rather than numerical, relations. Many such reasoning tasks, e.g., Allen's interval algebra, can be solved in 2^O(n*log n) time, but single-exponential running times 2^O(n) are currently far out of reach. \r\nIn this paper we consider single-exponential algorithms via a multivariate analysis consisting of a fine-grained parameter n (e.g., the number of variables) and a coarse-grained parameter k expected to be relatively small.\r\nWe introduce the classes FPE and XE of problems solvable in f(k)*2^O(n), respectively f(k)^n, time, and prove several fundamental properties of these classes. We proceed by studying temporal reasoning problems and (1) show that the partially ordered time problem of effective width k is solvable in 16^{kn} time and is thus included in XE, and (2) that the network consistency problem for Allen's interval algebra with no interval overlapping with more than k others is solvable in (2nk)^{2k}*2^n time and is included in FPE. Our multivariate approach is in no way limited to these to specific problems and may be a generally useful approach for obtaining single-exponential algorithms.","doi":"10.24963/ijcai.2022/251","url":"https://ijcai.org/proceedings/2022/251","pdf":"https://ijcai.org/proceedings/2022/0251.pdf","markdown":"https://ijcai.org/proceedings/2022/251.md","bibtex":"https://ijcai.org/proceedings/2022/bibtex/251","volume":{"slug":"2022","title":"Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2022","url":"https://ijcai.org/proceedings/2022/","year":2022,"location":"Vienna","dates":"23-29 July 2022","editors":"Luc De Raedt","isbn":"978-1-956792-00-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2022"},"section":"Constraint Satisfaction and Optimization","track":"Main Track","number":251,"pages":{"first":1804,"last":1810},"keywords":["Constraint Satisfaction and Optimization: Constraint Satisfaction"],"videos":[{"label":"Video #1","url":"https://slideslive.com/38986711","length_sec":62,"provider":"slideslive"},{"label":"Video #2","url":"https://slideslive.com/38984854","length_sec":347,"provider":"slideslive"}],"references":[{"position":1,"raw":"[Allen and Koomen, 1983] James F. 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