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On the Decidability of Finding a Positive ILP-Instance in a Regular Set of ILP-Instances

  • Petra WolfEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)

Abstract

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).

Keywords

Deterministic finite automaton Regular languages Regular intersection emptiness problem Decidability Integer linear programming 

Notes

Acknowledgment

The author thanks Markus L. Schmid for proofreading and helpful discussions and is grateful to the anonymous reviewers for their suggestions. The author was partially supported by DFG (FE 560/9-1).

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Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  1. 1.FB 4 - Abteilung InformatikwissenschaftenUniversität TrierTrierGermany

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