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

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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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References

  1. Alilooee, A., Banerjee, A.: Powers of edge ideals of regularity three bipartite graphs. J. Commut. Algebra 9, 441–454 (2017)

    Article  MathSciNet  Google Scholar 

  2. Alilooee, A., Beyarslan, S., Selvaraja, S.: Regularity of powers of unicyclic graphs. Rocky Mountain J. Math. 49, 699–728 (2019)

    Article  MathSciNet  Google Scholar 

  3. Banerjee, A.: The regularity of powers of edge ideals. J. Algebraic Combin. 41, 303–321 (2015)

    Article  MathSciNet  Google Scholar 

  4. Berlekamp, D.: Regularity defect stabilization of powers of an ideal. Math. Res. Lett. 19, 109–119 (2012)

    Article  MathSciNet  Google Scholar 

  5. Beyarslan, S., Ha, H.T., Trung, T.N.: Regularity of powers of forests and cycles. J. Algebraic Combin. 42, 1077–1095 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, pp. 1–48. Cambridge Univ. Press, Cambridge (1993)

  7. Brauer, A.: On a problem of partitions. Am. J. Math. 64, 299–312 (1942)

    Article  Google Scholar 

  8. Chardin, M.: Regularity stabilization for the powers of graded M-primary ideals. Proc. Am. Math. Soc. 143, 3343–3349 (2015)

    Article  MathSciNet  Google Scholar 

  9. Cutkosky, D., Herzog, J., Trung, N.V.: Asymptotic behavior of the Castelnuovo-Mumford regularity. Compositio Math. 118, 243–261 (1999)

    Article  MathSciNet  Google Scholar 

  10. Dung, L.X., Hien, T.T., Nguyen, Hop D., Trung, T.N.: Regularity and Koszul property of symbolic powers of monomial ideals. arXiv:1903.09026

  11. Eisenbud, D.: The Geometry of Syzygies. A Second Course in Commutative Algebra and Algebraic Geometry. Graduate Texts in Mathematics, vol. 229. Springer, New York (2005)

    MATH  Google Scholar 

  12. Eisenbud, D., Ulrich, B.: Notes on regularity stabilization. Proc. Am. Math. Soc. 140, 1221–1232 (2012)

    Article  MathSciNet  Google Scholar 

  13. Ehrhart, E.: Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux. J. Reine Angew. Math. 226, 1–29 (1967)

    MathSciNet  MATH  Google Scholar 

  14. Ehrhart, E.: Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires. J. Reine Angew. Math. 227, 25–49 (1967)

    MathSciNet  Google Scholar 

  15. Giang, D.H., Hoa, L.T.: On local cohomology of a tetrahedral curve. Acta Math. Vietnam. 35, 229–241 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)

    Article  MathSciNet  Google Scholar 

  17. Hang, N.T., Trung, T.N.: Regularity of powers of cover ideals of unimodular hypergraphs. J. Algebra 513, 159–176 (2018)

    Article  MathSciNet  Google Scholar 

  18. Herzog, J., Hoa, L.T., Trung, N.V.: Asymptotic linear bounds for the Castelnuovo–Mumford regularity. Trans. Am. Math. Soc. 354, 1793–1809 (2002)

    Article  MathSciNet  Google Scholar 

  19. Hoa, L.T., Trung, T.N.: Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals. Math. Proc. Camb. Philos. Soc. 149, 1–18 (2010)

    Article  MathSciNet  Google Scholar 

  20. Hoa, L.T., Trung, T.N.: Castelnuovo–Mumford regularity of symbolic powers of two-dimensional square-free monomial ideals. J. Commun. Algebra 8, 77–88 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Hoa, L.T., Trung, T.N.: Stability of depth and Cohen-Macaulayness of integral closures of powers of monomial ideals. Acta Math. Vietnam. 43, 67–81 (2018)

    Article  MathSciNet  Google Scholar 

  22. Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Combin. 47, 17–38 (2018)

    Article  MathSciNet  Google Scholar 

  23. Jayanthan, A.V., Selvaraja, S.: Asymptotic behavior of Castelnuovo–Mumford regularity of edge ideals of very well-covered graphs. arXiv:1708.06883

  24. Kodiyalam, V.: Asymptotic behaviour of Castelnuovo–Mumford regularity. Proc. Am. Math. Soc. 128, 407–411 (2000)

    Article  MathSciNet  Google Scholar 

  25. Minh, N.C., Trung, N.V.: Cohen–Macaulayness of powers of two-dimensional squarefree monomial ideals. J. Algebra 322, 4219–4227 (2009)

    Article  MathSciNet  Google Scholar 

  26. Reid, L., Roberts, L.G., Vitulli, M.A.: Some results on normal homogeneous ideals. Commun. Algebra 31, 4485–4506 (2003)

    Article  MathSciNet  Google Scholar 

  27. Romeijnders, W., Schultz, R., van der Vlerk, M.H., Klein Haneveld, W.K.: A convex approximation for two-stage mixed-integer recourse models with a uniform error bound. SIAM J. Optim. 26, 426–447 (2016)

    Article  MathSciNet  Google Scholar 

  28. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (2000)

    MATH  Google Scholar 

  29. Shen, B.: Parametrizing an integer linear program by an integer. SIAM J. Discrete Math. 32, 173–190 (2018)

    Article  MathSciNet  Google Scholar 

  30. Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston Inc, Boston (1996)

    Google Scholar 

  31. Sturmfels, B., Thomas, R.R.: Variation of cost functions in integer programming. Math. Programm. 77, 357–387 (1997). Ser. A

    MathSciNet  MATH  Google Scholar 

  32. Takayama, Y.: Combinatorial characterizations of generalized Cohen–Macaulay monomial ideals. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48, 327–344 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Trung, T.N.: Stability of associated primes of integral closures of monomial ideals. J. Combin. Ser. A. 116, 44–54 (2009)

    Article  MathSciNet  Google Scholar 

  34. Walkup, D.W., Wets, R.J.-B.: Lifting projections of convex polyhedra. Pacific J. Math. 28, 465–475 (1969)

    Article  MathSciNet  Google Scholar 

  35. Wolsey, L.A.: The b-hull of an integer program. Discrete Appl. Math. 3, 193–201 (1981)

    Article  MathSciNet  Google Scholar 

  36. Woods, K.: The unreasonable ubiquitousness of quasi-polynomials. Electron. J. Combin. 21, (2014). Paper 1.44, 23 pp

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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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  1. Institute of Mathematics, VAST, 18 Hoang Quoc Viet, 10307, Hanoi, Viet Nam

    Le Tuan Hoa

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  1. Le Tuan Hoa
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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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