On Proximity for k-Regular Mixed-Integer Linear Optimization

  • Luze Xu
  • Jon LeeEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


Putting a finer structure on a constraint matrix than is afforded by subdeterminant bounds, we give sharpened proximity results for the setting of k-regular mixed-integer linear optimization.


Mixed-integer linear optimization Proximity k-regular 


  1. 1.
    Aliev, I., Henk, M., Oertel, T.: Distances to lattice points in knapsack polyhedra. arXiv preprint arXiv:1805.04592 (2018)
  2. 2.
    Cook, W., Gerards, A.M., Schrijver, A., Tardos, É.: Sensitivity theorems in integer linear programming. Math. Program. 34(3), 251–264 (1986)Google Scholar
  3. 3.
    Eisenbrand, F., Weismantel, R.: Proximity results and faster algorithms for integer programming using the Steinitz lemma. In: SODA. pp. 808–816 (2018)Google Scholar
  4. 4.
    Granot, F., Skorin-Kapov, J.: Some proximity and sensitivity results in quadratic integer programming. Math. Program. 47(1–3), 259–268 (1990)Google Scholar
  5. 5.
    Hochbaum, D.S., Shanthikumar, J.G.: Convex separable optimization is not much harder than linear optimization. J. ACM 37(4), 843–862 (1990)Google Scholar
  6. 6.
    Lee, J.: Subspaces with well-scaled frames. Ph.D. dissertation, Cornell University (1986)Google Scholar
  7. 7.
    Lee, J.: Subspaces with well-scaled frames. Linear Algebra Appl. 114, 21–56 (1989)Google Scholar
  8. 8.
    Lee, J.: The incidence structure of subspaces with well-scaled frames. J. Comb. Theory Ser. B 50(2), 265–287 (1990)Google Scholar
  9. 9.
    Paat, J., Weismantel, R., Weltge, S.: Distances between optimal solutions of mixed-integer programs. Math. Program. (2018)
  10. 10.
    Rockafellar, R.T.: The elementary vectors of a subspace of \({\mathbb{R}}^n\). In: Combinatorial Mathematics and Its Applications, pp. 104–127. University of North Carolina Press (1969)Google Scholar
  11. 11.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley (1998)Google Scholar
  12. 12.
    Veselov, S.I., Chirkov, A.J.: Integer program with bimodular matrix. Discrete Optim. 6(2), 220–222 (2009)Google Scholar
  13. 13.
    Werman, M., Magagnosc, D.: The relationship between integer and real solutions of constrained convex programming. Math. Prog. 51(1), 133–135 (1991)Google Scholar
  14. 14.
    Zaslavsky, T.: Signed graphs. Discrete Appl. Math. 4(1), 47–74 (1982)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA

Personalised recommendations