{"type":"ScholarlyArticle","title":"Unary Integer Linear Programming with Structural Restrictions","authors":[{"name":"Eduard Eiben","slug":"eduard-eiben","given":"Eduard","family":"Eiben","affiliation":"Department of Informatics , University of Bergen , Norway"},{"name":"Robert Ganian","slug":"robert-ganian","given":"Robert","family":"Ganian","affiliation":"Algorithms and Complexity group , TU Wien , Austria"},{"name":"Dušan Knop","slug":"dusan-knop","given":"Dušan","family":"Knop","affiliation":"Department of Informatics , University of Bergen , Norway"},{"name":"Sebastian Ordyniak","slug":"sebastian-ordyniak","given":"Sebastian","family":"Ordyniak","affiliation":"Algorithms group , University of Sheffield , UK"}],"abstract":"Recently a number of algorithmic results have appeared which show the tractability of Integer Linear Programming (ILP) instances under strong restrictions on variable domains and/or coefficients (AAAI 2016, AAAI 2017, IJCAI 2017). In this paper, we target ILPs where neither the variable domains nor the coefficients are restricted by a fixed constant or parameter; instead, we only require that our instances can be encoded in unary.\r\n\r\nWe provide new algorithms and lower bounds for such ILPs by exploiting the structure of their variable interactions, represented as a graph. Our first set of results focuses on solving ILP instances through the use of a graph parameter called clique-width, which can be seen as an extension of treewidth which also captures well-structured dense graphs. In particular, we obtain a polynomial-time algorithm for instances of bounded clique-width whose domain and coefficients are polynomially bounded by the input size, and we complement this positive result by a number of algorithmic lower bounds. Afterwards, we turn our attention to ILPs with acyclic variable interactions. In this setting, we obtain a complexity map for the problem with respect to the graph representation used and restrictions on the encoding.","doi":"10.24963/ijcai.2018/179","url":"https://ijcai.org/proceedings/2018/179","pdf":"https://ijcai.org/proceedings/2018/0179.pdf","markdown":"https://ijcai.org/proceedings/2018/179.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/179","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Constraints and SAT","track":"Main track","number":179,"pages":{"first":1284,"last":1290},"keywords":["Constraints and SAT: Dynamic Programming","Heuristic Search and Game Playing: Combinatorial Search and Optimisation","Constraints and SAT: Constraint Optimisation"],"videos":[],"references":[{"position":1,"raw":"[Alumur and Kara, 2008] Sibel A. 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