Hilbert basis (linear programming)

In linear programming, a Hilbert basis for a convex cone C is an integer cone basis: minimal set of integer vectors such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.

Definition

A set $A=\{a_{1},\ldots ,a_{n}\}$ of integer vectors is a Hilbert basis of its convex cone

$C=\{\lambda _{1}a_{1}+\ldots +\lambda _{n}a_{n}\mid \lambda _{1},\ldots ,\lambda _{n}\geq 0,\lambda _{1},\ldots ,\lambda _{n}\in {\mathbb {R}}\}$

if every integer vector from C belongs to the integer convex cone of A:

$\{\alpha _{1}a_{1}+\ldots +\alpha _{n}a_{n}\mid \alpha _{1},\ldots ,\alpha _{n}\geq 0,\alpha _{1},\ldots ,\alpha _{n}\in {\mathbb {Z}}\},$

and no vector from A belongs to the integer convex cone of the others.

References

• Bruns, Winfried; Gubeladze, Joseph; Henk, Martin; Martin, Alexander; Weismantel, Robert (1999), "A counterexample to an integer analogue of Carathéodory's theorem", Journal für die reine und angewandte Mathematik, 510 (510): 179–185, doi:10.1515/crll.1999.045
• Cook, William John; Fonlupt, Jean; Schrijver, Alexander (1986), "An integer analogue of Carathéodory's theorem", Journal of Combinatorial Theory. Series B, 40 (1): 63–70, doi:10.1016/0095-8956(86)90064-X
• Eisenbrand, Friedrich; Shmonin, Gennady (2006), "Carathéodory bounds for integer cones", Operations Research Letters, 34 (5): 564–568, doi:10.1016/j.orl.2005.09.008
• D. V. Pasechnik (2001). "On computing the Hilbert bases via the Elliott—MacMahon algorithm". Theoretical Computer Science. 263: 37–46. doi:10.1016/S0304-3975(00)00229-2.