Hilbert basis (linear programming)

The Hilbert basis of a convex cone C is a minimal set of integer vectors in C such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.

Definition

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Hilbert basis visualization. Two rays in the plane define an infinite cone of all the points lying between them. The unique Hilbert basis points of the cone are circled in yellow. Every integer point in the cone can be written as a sum of these basis elements. As you change the cone by moving one of the rays, the Hilbert basis also changes.

Given a lattice and a convex polyhedral cone with generators

we consider the monoid . By Gordan's lemma, this monoid is finitely generated, i.e., there exists a finite set of lattice points such that every lattice point is an integer conical combination of these points:

The cone C is called pointed if implies . In this case there exists a unique minimal generating set of the monoid —the Hilbert basis of C. It is given by the set of irreducible lattice points: An element is called irreducible if it can not be written as the sum of two non-zero elements, i.e., implies or .

References

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  • 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. hdl:10220/8240.