A faster algorithm for counting the integer points number in $\Delta$-modular polyhedra (corrected version)

TL;DR

This paper introduces a new fixed-parameter tractable algorithm for counting integer points in $\Delta$-modular polyhedra, improving efficiency over previous methods by avoiding Barvinok's decomposition and using exponential series representations.

Contribution

The authors develop a novel FPT-algorithm for counting integer points in $\Delta$-modular polyhedra that surpasses existing approaches in efficiency and does not rely on Barvinok's unimodular sign decomposition.

Findings

The new algorithm is more efficient for $\Delta$-modular problems.

It provides FPT algorithms for counting points in $\Delta$-modular simplices.

It enables fixed-parameter tractable solutions for $\Delta$-modular subset-sum problems.

Abstract

Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $\Delta$-modular, where the polytope $P$ is called $\Delta$-modular if all the rank sub-determinants of $A$ are bounded by $\Delta$ in the absolute value. We present a new FPT-algorithm, parameterized by $\Delta$ and by the maximal number of vertices in $P$, where the maximum is taken by all r.h.s. vectors $b$. We show that our algorithm is more efficient for $\Delta$-modular problems than the approach of A. Barvinok et al. To this end, we do not directly compute the short rational generating function for $P \cap Z^n$, which is commonly used for the considered problem. Instead, we use the dynamic programming principle to compute its particular representation in the form of exponential series that depends on a single variable. We completely do not rely to the Barvinok's unimodular sign decomposition technique. Using our new complexity bound, we consider different special cases that may be of independent interest. For example, we give FPT-algorithms for counting the integer points number in $\Delta$-modular simplices and similar polytopes that have $n + O(1)$ facets. As a special case, for any fixed $m$, we give an FPT-algorithm to count solutions of the unbounded $m$-dimensional $\Delta$-modular subset-sum problem.