Hilbert basis (linear programming)

The Hilbert basis of a convex cone C is a 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

Hilbert basis visualization

Given a lattice ${\displaystyle L\subset \mathbb {Z} ^{d}}$ and a convex polyhedral cone with generators ${\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {Z} ^{d}}$

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

we consider the monoid ${\displaystyle C\cap L}$. By Gordan's lemma this monoid is finitely generated, i.e., there exists a finite set of lattice points ${\displaystyle \{x_{1},\ldots ,x_{m}\}\subset C\cap L}$ such that every lattice point ${\displaystyle x\in C\cap L}$ is an integer conical combination of these points:

${\displaystyle x=\lambda _{1}x_{1}+\ldots +\lambda _{m}x_{m},\quad \lambda _{1},\ldots ,\lambda _{m}\in \mathbb {Z} ,\lambda _{1},\ldots ,\lambda _{m}\geq 0}$

The cone C is called pointed, if ${\displaystyle x,-x\in C}$ implies ${\displaystyle x=0}$. In this case there exists a unique minimal generating set of the monoid ${\displaystyle C\cap L}$ - the Hilbert basis of C. It is given by set of irreducible lattice points: An element ${\displaystyle x\in C\cap L}$ is called irreducible if it can not be written as the sum of two non-zero elements, i.e., ${\displaystyle x=y+z}$ implies ${\displaystyle y=0}$ or ${\displaystyle z=0}$.

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, 1999 (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 (1–2): 37–46. doi:10.1016/S0304-3975(00)00229-2.