{"type":"ScholarlyArticle","title":"Solving Integer Linear Programs with a Small Number of Global Variables and Constraints","authors":[{"name":"Pavel Dvořák","slug":"pavel-dvorak","given":"Pavel","family":"Dvořák","affiliation":"Charles University , Prague , Czech Republic"},{"name":"Eduard Eiben","slug":"eduard-eiben","given":"Eduard","family":"Eiben","affiliation":"TU Wien , Vienna , Austria"},{"name":"Robert Ganian","slug":"robert-ganian","given":"Robert","family":"Ganian","affiliation":"TU Wien , Vienna , Austria"},{"name":"Dušan Knop","slug":"dusan-knop","given":"Dušan","family":"Knop","affiliation":"Charles University , Prague , Czech Republic"},{"name":"Sebastian Ordyniak","slug":"sebastian-ordyniak","given":"Sebastian","family":"Ordyniak","affiliation":"TU Wien , Vienna , Austria"}],"abstract":"Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of ``global'' variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N-fold ILP instances. \r\n\r\nOur main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.","doi":"10.24963/ijcai.2017/85","url":"https://ijcai.org/proceedings/2017/85","pdf":"https://ijcai.org/proceedings/2017/0085.pdf","markdown":"https://ijcai.org/proceedings/2017/85.md","bibtex":"https://ijcai.org/proceedings/2017/bibtex/85","volume":{"slug":"2017","title":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2017","url":"https://ijcai.org/proceedings/2017/","year":2017,"location":"Melbourne, Australia","dates":"19-25 August 2017","editors":"Carles Sierra","isbn":"978-0-9992411-0-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2017"},"section":"Constraints and Satisfiability","track":"Main track","number":85,"pages":{"first":607,"last":613},"keywords":["Constraints and Satisfiability: Constraints and Satisfiability","Combinatorial & Heuristic Search: Combinatorial search/optimisation","Constraints and Satisfiability: Constraint Optimisation","Constraints and Satisfiability: MaxSAT, MinSAT"],"videos":[],"references":[{"position":1,"raw":"[Aigner et al., 2010] Martin Aigner, Günter M Ziegler, and Alfio Quarteroni. 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