On the decidability of finding a positive ILP-instance in a regular set of ILP-instances

The regular intersection emptiness problem for a decision problem P (\({{\textit{int}}_{{\mathrm {Reg}}}}\)(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since \({{\textit{int}}_{{\mathrm {Reg}}}}\)(P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the \({{\textit{int}}_{{\mathrm {Reg}}}}\)-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the \({{\textit{int}}_{{\mathrm {Reg}}}}\)-problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of \({{\textit{int}}_{{\mathrm {Reg}}}}\)(ILP).