Hilbert basis (linear programming): Difference between revisions

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The '''Hilbert basis''' of a [[convex cone]] ''C'' is a minimal set of integer [[vector (mathematics)|vector]]s in ''C'' such that every [[integer]] vector in ''C'' is a [[conical combination]] of the vectors in the Hilbert basis with integer coefficients.

== Definition ==

[[File:Hilbert basis.gif|thumb|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 (group)|lattice]] <math>L\subset\mathbb{Z}^nd</math> and a convex polyhedral cone with generators <math>a_1,\ldots,a_n\in\mathbb{Z}^d</math>

:<math>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,</math>

Given a [[Lattice (group)|lattice]] <math>L\subset\mathbb{Z}^n</math> and a convex polyhedral cone

we consider the [[monoid]] <math>C\cap L</math>. By [[Gordan's lemma]], this monoid is finetelyfinitely generated, i.e., there exists a [[finite set]] of lattice points <math>\{x_1,\ldots,x_m\}\subset C\cap L</math> such that every lattice point <math>x\in C\cap L</math> is an integer conical combination of these points:

:<math>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}\}</math>

:<math> x=\lambda_1 x_1+\ldots+\lambda_m x_m, \quad\lambda_1,\ldots,\lambda_m\in\mathbb{Z}, \lambda_1,\ldots,\lambda_m\geq0.</math>

we consider the [[monoid]] <math>C\cap L</math>. By [[Gordan's lemma]] this monoid is finetely generated, i.e., there exists a finite set of lattice points <math>\{x_1,\ldots,x_m\}\subset C\cap L</math> such that every lattice point <math>x\in C\cap L</math> is an integer conical combination of these points:

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

:<math> x=\lambda_1 x_1+\ldots+\lambda_m x_m, \quad\lambda_1,\ldots,\lambda_m\in\mathbb{Z}, \lambda_1,\ldots,\lambda_m\geq0</math>

The '''Hilbert basis''' of ''C'' is the minimal generating set of the monoid <math>C\cap L</math>. It is given by set of irreducible lattice points: An element <math>x\in C\cap L</math> is called irreducible if it can not be written as the sum of two non-zero elements, i.e., <math>x=y+z</math> implies <math>y=0</math> or <math>z=0</math>.

== References ==

* {{Citation | last1=Bruns | first1=Winfried |last2=Gubeladze | first2=Joseph | last3=Henk | first3=Martin | last4=Martin | first4=Alexander | last5=Weismantel | first5=Robert | title=A counterexample to an integer analogue of Carathéodory's theorem | doi=10.1515/crll.1999.045 | year=1999 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=5101999 | pages=179–185 | issue=510}}

* {{Citation | last1=Cook | first1=William John | last2=Fonlupt | first2=Jean | last3=Schrijver | first3=Alexander | authorlink3=Alexander Schrijver|title=An integer analogue of Carathéodory's theorem | doi=10.1016/0095-8956(86)90064-X | year=1986 | journal=Journal of Combinatorial Theory., Series B | volume=40 | issue=1 | pages=63–70| doi-access=free }}

* {{Citation | last1=Eisenbrand | first1=Friedrich | last2=Shmonin | first2=Gennady | title=Carathéodory bounds for integer cones | doi=10.1016/j.orl.2005.09.008 | year=2006 | journal=Operations Research Letters | volume=34 | issue=5 | pages=564–568| url=http://infoscience.epfl.ch/record/121640 }}

* {{cite journal |author = D. V. Pasechnik |title=On computing the Hilbert bases via the Elliott—MacMahon algorithm |journal=Theoretical Computer Science |volume=263 |issue=1–2 |year=2001 |pages=37–46 |doi=10.1016/S0304-3975(00)00229-2|doi-access=free |hdl=10220/8240 |hdl-access=free }}

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