• 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}
}
• Published 4 October 2021
• Computer Science
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
• Mathematics
ArXiv
• 2020
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
• Mathematics
• 2002
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
• Mathematics
Math. Oper. Res.
• 2020
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
• Mathematics
J. Symb. Comput.
• 2008
A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
• Mathematics
SCG '91
• 1991
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
• Mathematics
Algorithmica
• 2006
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
• Mathematics
Comb.
• 1992
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
• Mathematics
Journal of Global Optimization
• 2022
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
• Mathematics, Computer Science
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
• 2011
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).