Integer Knapsacks: Average Behavior of the Frobenius Numbers

Published Online:https://doi.org/10.1287/moor.1090.0393

References

  • Aardal K., Lenstra A. Hard equality constrained integer knapsacks. Math. Oper. Res. (2004) 29(3):724–738LinkGoogle Scholar
  • Aliev I., Gruber P. M. An optimal lower bound for the Frobenius problem. J. Number Theory (2007) 123(1):71–79CrossrefGoogle Scholar
  • Arnold V. I. Weak asymptotics of the numbers of solutions of Diophantine equations. Funktsionalnyi. Anal. i Prilozhen (1999) 33(4):65–66CrossrefGoogle Scholar
  • Arnold V. I. Geometry and growth rate of Frobenius numbers of additive semigroups. Math. Phys. Anal. Geometry (2006) 9(2):95–108CrossrefGoogle Scholar
  • Arnold V. I. Arithmetical turbulence of selfsimilar fluctuations statistics of large Frobenius numbers of additive semigroups of integers. Moscow Math. J. (2007) 7(2):173–193CrossrefGoogle Scholar
  • Beck M., Zacks S. Refined upper bounds for the linear Diophantine problem of Frobenius. Adv. Appl. Math. (2004) 32(3):454–467CrossrefGoogle Scholar
  • Beck M., Diaz R., Robins S. The Frobenius problem, rational polytopes, and Fourier-Dedekind sums. J. Number Theory (2002) 96(1):1–21Google Scholar
  • Beihoffer D., Hendry J., Nijenhuis A., Wagon S. Faster algorithms for Frobenius numbers. Electronic J. Combin. (2005) 12(Research Paper 27), 38 (electronic). http://www.combinatorics.orgGoogle Scholar
  • Bourgain J., Sinaĭ Ya. G. Limit behavior of large Frobenius numbers. Uspekhi Matematicheskikh Nauk (2007) 62(4, 376):77–90CrossrefGoogle Scholar
  • Davison J. L. On the linear Diophantine problem of Frobenius. J. Number Theory (1994) 48(3):353–363CrossrefGoogle Scholar
  • Einstein D., Lichtblau D., Strzebonski A., Wagon S. Frobenius numbers by lattice point enumeration. Integers (2007) 7(15):63Google Scholar
  • Erdős P., Graham R. On a linear Diophantine problem of Frobenius. Acta Arithmetica (1972) 21:399–408CrossrefGoogle Scholar
  • Fukshansky L., Robins S. Frobenius problem and the covering radius of a lattice. Discrete Comput. Geometry (2007) 37(3):471–483CrossrefGoogle Scholar
  • Gruber P. M.Convex and Discrete Geometry (2007) (Springer, Berlin) Google Scholar
  • Gruber P. M., Lekkerkerker C. G.Geometry of Numbers (1987) (North–Holland, Amsterdam) Google Scholar
  • Hansen P., Ryan J. Testing integer knapsacks for feasibility. Eur. J. Oper. Res. (1996) 88(3):578–582CrossrefGoogle Scholar
  • Kannan R. Lattice translates of a polytope and the Frobenius problem. Combinatorica (1992) 12(2):161–177CrossrefGoogle Scholar
  • Kannan R., Lovász L. Covering minima and lattice-point-free convex bodies. Ann. Math. (2) (1988) 128(3):577–602CrossrefGoogle Scholar
  • Karp R. M., Miller R. E., Thatcher J. W. Reducibility among combinatorial problems. Complexity of Computer Computations (1972) (Plenum, New York) 85–103CrossrefGoogle Scholar
  • Lee J., Onn S., Weismantel R., Goldberg A., Zhou Y. Nonlinear optimization over a weighted independence system. Proc. 2009 AAIM (2009) 5564(Springer, Berlin) 251–264LNCSCrossrefGoogle Scholar
  • Marklof J. The asymptotic distribution of Frobenius numbers. (2009) . arXiv.org. arXiv:0902.3557Google Scholar
  • Ramírez Alfonsín J. L. Complexity of the Frobenius problem. Combinatorica (1996) 16(1):143–147CrossrefGoogle Scholar
  • Ramírez Alfonsín J. L. The Diophantine Frobenius problem. Oxford Lecture Series in Mathematics and Its Applications (2005) (Oxford University Press, Oxford, UK) CrossrefGoogle Scholar
  • Schlage-Puchta J.-C. An estimate for Frobenius' Diophantine problem in three dimensions. J. Integer Sequences (2005) 8(1, Article 05.1.7) 4 (electronic). http://www.cs.uwaterloo.ca/journals/JISGoogle Scholar
  • Schmidt W. M. The distribution of sublattices of Zm. Monatshefte Math. (1998) 125(1):37–81CrossrefGoogle Scholar
  • Selmer E. On the linear Diophantine problem of Frobenius. J. Reine Angewandte Math. (1977) 293/294:1–17Google Scholar
  • Shur V., Sinaĭ Ya. G., Ustinov A. Limiting distribution of Frobenius numbers for n = 3. (2009) . arXiv.org. arXiv:0810.5219v1Google Scholar
  • Ustinov A. On the distribution of Frobenius numbers with three arguments I. Mat. Sb. (2009) . ForthcomingCrossrefGoogle Scholar
  • Vitek Y. Bounds for a linear Diophantine problem of Frobenius. J. London Math. Soc. (1975) 10(2):79–85CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.