TY - JOUR

T1 - Computing with an algebraic-perturbation variant of Barvinok's algorithm

AU - Lee, Jon

AU - Skipper, Daphne

N1 - Funding Information:
The authors gratefully acknowledge Jesús De Loera for his many answers produced under relentless questioning. J. Lee was partially supported by NSF grant CMMI-1160915 and ONR grant N00014-14-1-0315 .

PY - 2018/5/11

Y1 - 2018/5/11

N2 - We present a variant of Barvinok's algorithm for computing a short rational generating function for the integer points in a nonempty pointed polyhedron P:={x∈Rn:Ax≤b} given by rational inequalities. A main use of such a rational generating function is to count the number of integer points in P. Our variant is based on making an algebraic perturbation of the right-hand side b∈Qm of the inequalities, replacing each bi with bi+τi, where τ is considered to be an arbitrarily small positive real indeterminate. Hence, elements of our right-hand side vector become elements of the ordered ring Q[τ] of polynomials in τ. Denoting the algebraically-perturbed polyhedron as P(τ)⊂R[τ]n, we readily see that: (i) P(τ) is always full dimensional, (ii) P(τ) is always simple, and (iii) P(τ) contains the same integer points as P. Unlike other versions of Barvinok's algorithm, we will have to do some arithmetic in Q[τ]. However, because of (i) we will not need to preprocess our input polyhedron if it is not full dimensional, and because of (ii) we will not need to triangulate tangent cones at the vertices of the polyhedron. We give the details of our perturbation variant of Barvinok's algorithm, describe a proof-of-concept implementation developed in Mathematica, and present results of computational experiments.

AB - We present a variant of Barvinok's algorithm for computing a short rational generating function for the integer points in a nonempty pointed polyhedron P:={x∈Rn:Ax≤b} given by rational inequalities. A main use of such a rational generating function is to count the number of integer points in P. Our variant is based on making an algebraic perturbation of the right-hand side b∈Qm of the inequalities, replacing each bi with bi+τi, where τ is considered to be an arbitrarily small positive real indeterminate. Hence, elements of our right-hand side vector become elements of the ordered ring Q[τ] of polynomials in τ. Denoting the algebraically-perturbed polyhedron as P(τ)⊂R[τ]n, we readily see that: (i) P(τ) is always full dimensional, (ii) P(τ) is always simple, and (iii) P(τ) contains the same integer points as P. Unlike other versions of Barvinok's algorithm, we will have to do some arithmetic in Q[τ]. However, because of (i) we will not need to preprocess our input polyhedron if it is not full dimensional, and because of (ii) we will not need to triangulate tangent cones at the vertices of the polyhedron. We give the details of our perturbation variant of Barvinok's algorithm, describe a proof-of-concept implementation developed in Mathematica, and present results of computational experiments.

KW - Barvinok's algorithm

KW - Polytope

KW - Rational generating function

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U2 - 10.1016/j.dam.2015.12.003

DO - 10.1016/j.dam.2015.12.003

M3 - Article

AN - SCOPUS:84973103286

VL - 240

SP - 63

EP - 77

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -