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 (\( int_{\mathrm {Reg}} \)(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since \( int_{\mathrm {Reg}} \)(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 \( int_{\mathrm {Reg}} \)-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 \( int_{\mathrm {Reg}} \)-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 \( int_{\mathrm {Reg}} \)(ILP).

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