Integer Linear Programming (ILP) can be seen as the archetypical problem for NPcomplete 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.
Ganian and Ordyniak REF showed fixed-parameter tractability for the combined parameter the treedepth and the largest absolute value in the constraint matrix, and contrasted this with a W[1]-hardness result when treedepth is exchanged for treewidth.
The Complexity Landscape of Decompositional Parameters for ILP
ArXiv
ArXiv
Mathematics Computer Science