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Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity

Abstract

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n. An integer \(N_*\) is determined such that the optima of these integer programs are a quasi-linear function of n for all \(n\ge N_*\). Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo–Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.

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Acknowledgements

The author would like to thank professors Martin Grötschel, Jesus De Loera, Alexander Barvinok and Dr. Hoang Nam Dung for their consultation on Linear and Integer Programming. In particular, thanks to them I am aware of references [29, 31, 35, 36], and the proof of Proposition 1.3 was simplified. The author also would like to thank both referees for their critical comments and useful suggestions, especially for pointing out the reference [16]. This work is partially supported by NAFOSTED (Vietnam) under the grant number 101.04-2018.307.

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Correspondence to Le Tuan Hoa.

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Dedicated to Professor Ngo Viet Trung on the occasion of his 65th birthday.

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Hoa, L.T. Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity. Math. Program. 193, 157–194 (2022). https://doi.org/10.1007/s10107-020-01595-x

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  • DOI: https://doi.org/10.1007/s10107-020-01595-x

Keywords

  • Linear Programming
  • Integer Programming
  • Monomial ideal
  • Integral closure
  • Castelnuovo–Mumford regularity

Mathematics Subject Classification

  • 13D45
  • 90C10