• 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}
}

• D. Gribanov, D. Malyshev
• 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 a d-dimensional polytope P, defined by one of the systems above. Using the obtained result, we have the following arithmetical complexity bounds to compute | P ∩Z |: – The bound O( d m+1) m ·d ·∆ ·log(∆) that is polynomial in d and∆, for any fixed m. Takingm = 1, it gives an O ( d ·∆ ·log(∆) ) -algorithm to compute the number of integer points in a simplex or the number of solutions of the unbounded Subset-Sum problem, where ∆ means the maximal weight of an item. The article was prepared under financial support of Russian Science Foundation grant No 21-11-00194. D. V. Gribanov National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod, 603155, Russian Federation E-mail: [e-mail] D. S. Malyshev National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod, 603155, Russian Federation E-mail: [e-mail] 2 D. V. Gribanov, D. S. Malyshev – The bound O ( m d +1 ) d 2 · d ·∆ · log(∆) that is polynomial in m and ∆, for any fixed d. The last bound can be used to obtain a faster algorithm for the ILP feasibility problem, when the parameters m and ∆ are relatively small. For example, taking m = O(d) and ∆ = 2, the above bound becomes 2, which is faster, than the state of the art algorithm, due to [11,12] with the complexity bound O(d) · poly(φ), where φ = size(A, b); – The bound O(d) d 2 ·∆ · log(∆) that is polynomial in ∆, for any fixed d. Taking ∆ = O(1), the last bound becomes O(d) d 2 , which again gives a faster algorithm for the ILP feasibility problem, than the state of the art algorithm, due to [11,12]. The unbounded and parametric versions of the above problem are also considered.