The complexity landscape of decompositional parameters for ILP

Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are kno...

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Bibliographic Details
Published in:Artificial intelligence 2018-04, Vol.257, p.61-71
Main Authors: Ganian, Robert, Ordyniak, Sebastian
Format: Article
Language:English
Subjects:
(Parameterized) complexity
Artificial intelligence
Complexity
Complexity theory
Decomposition
Fragments
Integer Linear Programming
Integer programming
Parameter optimization
Parameters
Treedepth
Treewidth
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Summary:Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are known, probably the most prominent of which is based on the notion of total unimodularity. Using entirely different techniques, we identify new tractable fragments of ILP by studying structural parameterizations of the constraint matrix within the framework of parameterized complexity. In particular, we show that ILP is fixed-parameter tractable when parameterized by the treedepth of the constraint matrix and the maximum absolute value of any coefficient occurring in the ILP instance. Together with matching hardness results for the more general parameter treewidth, we give an overview of the complexity of ILP w.r.t. decompositional parameters defined on the constraint matrix.
ISSN:0004-3702
1872-7921
DOI:10.1016/j.artint.2017.12.006