Solving Integer Linear Programs with
a Small Number of Global Variables and Constraints
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.
Our 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