• Corpus ID: 238354202

Faster algorithm for counting of the integer points number in $\Delta$-modular polyhedra

@inproceedings{Gribanov2021FasterAF,
  title={Faster algorithm for counting of the integer points number in \$\Delta\$-modular polyhedra},
  author={Dmitry V. Gribanov and Dmitriy S. Malyshev},
  year={2021}
}
Let a polytope P be defined by one of the following ways: (i) P = {x ∈ R : Ax ≤ b}, where A ∈ Z, b ∈ Q, rank(A) = n and d := dim(P) = n; (ii) P = {x ∈ R+ : Ax = b}, where A ∈ Z , b ∈ Z, rank(A) = m and d := dim(P) = n−m; and let all the rank-order sub-determinants of A be bounded by ∆ in the absolute values. We show that | P ∩Z | can be computed with an algorithm, having the arithmetic complexity bound O ( ν(d,m,∆) · d ·∆ · log(∆) ) , where ν(d,m,∆) is the maximal possible number of vertices in… 

References

SHOWING 1-10 OF 54 REFERENCES
On lattice point counting in Δ-modular polyhedra
TLDR
The step polynomial representation of the function c_P(y) , where $P_{y}$ is parametric polytope, can be computed by aPolynomial time even in varying dimension if $P_y$ has a close structure to the cases (i) or (ii).
Short rational generating functions for lattice point problems
Abstract. We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we
Counting Integral Points in Polytopes via Numerical Analysis of Contour Integration
TLDR
A new type of an inclusion-exclusion formula for integer points in $P(y)$ is established, based on the standard error analysis to the numerical contour integration for the inverse Z-transform, and this improves, in terms of space complexity, a naive DP algorithm with $O((\|y\|_\infty + 1)^n)$-size DP table.
Lattice invariant valuations on rational polytopes
AbstractLetΛ be a lattice ind-dimensional euclidean space $$\mathbb{E}^d $$ , and $$\bar \Lambda $$ the rational vector space it generates. Ifϕ is a valuation invariant underΛ, andP is a polytope
Counting with rational generating functions
A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
AbstractWe present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the
Counting Integer Points in Parametric Polytopes Using Barvinok's Rational Functions
TLDR
This work extends an existing method, based on Barvinok's decomposition, for counting the number of integer points in a non-parametric polytope and computes polynomially-sized enumerators in polynomial time (for fixed dimensions).
On integer points in polyhedra
TLDR
An algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1+ε in time polynomial inm, ϕ and 1/ε when the dimensionn is fixed is described.
On $$\Delta $$-modular integer linear problems in the canonical form and equivalent problems
TLDR
This paper considers ILP problems in the canonical form max, where all the entries of A, b, c are integer, parameterized by the number of rows of A and ‖A‖max.
Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
TLDR
A novel algorithm for enumerating lattice points in any convex body known as the M-ellipsoid is given, and an expected O(f*(n))^n-time algorithm for Integer Programming, where f*( n) denotes the optimal bound in the so-calledflatnesstheorem, which is conjectured to be f* (n) = O(n).
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