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

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Descriptional Complexity of Formal Systems (DCFS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11612))

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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).

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Notes

  1. 1.

    A Dagstuhl seminar on ‘Graph Modification Problems’ was held in 2014 [2].

  2. 2.

    Note that this problem is only well-defined if it is clear how P is represented as a language, i. e., we have to define how P-instances are encoded as strings.

  3. 3.

    LOGSPACE and P also contain problems with undecidable \( int_{\mathrm {Reg}} \)-problem [23].

  4. 4.

    Our construction uses similar ideas as given in [11].

  5. 5.

    An extension \(\varvec{v'} \in \mathbb {Z}^n\) of \(\varvec{v} \in \mathbb {Z}^m\) with \(n >m\) coincides with \(\varvec{v}\) in all positions \(i \le m\).

  6. 6.

    The function \(\underset{{\text {lex}}}{\min }\) is used to make the definition clear. Any other element of the set could be used as well.

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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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Authors and Affiliations

  1. FB 4 - Abteilung Informatikwissenschaften, Universität Trier, Trier, Germany

    Petra Wolf

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  1. Petra Wolf
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Correspondence to Petra Wolf .

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Editors and Affiliations

  1. Slovak Academy of Sciences, Košice, Slovakia

    Michal Hospodár

  2. Slovak Academy of Sciences, Košice, Slovakia

    Galina Jirásková

  3. Saint Mary's University, Halifax, NS, Canada

    Stavros Konstantinidis

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Wolf, P. (2019). On the Decidability of Finding a Positive ILP-Instance in a Regular Set of ILP-Instances. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds) Descriptional Complexity of Formal Systems. DCFS 2019. Lecture Notes in Computer Science(), vol 11612. Springer, Cham. https://doi.org/10.1007/978-3-030-23247-4_21

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  • DOI: https://doi.org/10.1007/978-3-030-23247-4_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23246-7

  • Online ISBN: 978-3-030-23247-4

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