In our work, we consider the problem of computing a vector $x \in Z^n$ of minimum $\|\cdot\|_p$-norm such that $a^\top x \not= a_0$, for any vector $(a,a_0)$ from a given subset of $Z^n$ of size $m$. In other words, we search for a vector of minimum norm that avoids a given finite set of hyperplanes, which is natural to call as the $\textit{Hyperplanes Avoiding Problem}$. This problem naturally appears as a subproblem in Barvinok-type algorithms for counting integer points in polyhedra. We show that: 1) With respect to $\|\cdot\|_1$, the problem admits a feasible solution $x$ with $\|x\|_1 \leq (m+n)/2$, and show that such solution can be constructed by a deterministic polynomial-time algorithm with $O(n \cdot m)$ operations. Moreover, this inequality is the best possible. This is a significant improvement over the previous randomized algorithm, which computes $x$ with a guaranty $\|x\|_{1} \leq n \cdot m$. The original approach of A.~Barvinok can guarantee only $\|x\|_1 = O\bigl((n \cdot m)^n\bigr)$. To prove this result, we use a newly established algorithmic variant of the Combinatorial Nullstellensatz; 2) The problem is NP-hard with respect to any norm $\|\cdot\|_p$, for $p \in \bigl(R_{\geq 1} \cup \{\infty\}\bigr)$. 3) As an application, we show that the problem to count integer points in a polytope $P = \{x \in R^n \colon A x \leq b\}$, for given $A \in Z^{m \times n}$ and $b \in Q^m$, can be solved by an algorithm with $O\bigl(ν^2 \cdot n^3 \cdot Δ^3 \bigr)$ operations, where $ν$ is the maximum size of a normal fan triangulation of $P$, and $Δ$ is the maximum value of rank-order subdeterminants of $A$. As a further application, it provides a refined complexity bound for the counting problem in polyhedra of bounded codimension. For example, in the polyhedra of the Unbounded Subset-Sum problem.

翻译：在本文中，我们考虑如下问题：计算一个最小 $\|\cdot\|_p$-范数的向量 $x \in Z^n$，使得对于来自 $Z^n$ 中给定大小为 $m$ 的子集的任意向量 $(a,a_0)$，满足 $a^\top x \not= a_0$。换言之，我们寻找一个最小范数向量，它能规避给定的有限超平面集，这自然被称为 $\textit{超平面规避问题}$。该问题作为子问题自然地出现在用于多面体中整数点计数的Barvinok型算法中。我们证明：1）关于 $\|\cdot\|_1$，该问题存在一个可行解 $x$，满足 $\|x\|_1 \leq (m+n)/2$，并证明此类解可通过一个具有 $O(n \cdot m)$ 次操作的确定性多项式时间算法构造。此外，该不等式是最优的。这相较于之前保证 $\|x\|_{1} \leq n \cdot m$ 的随机算法有显著改进。A.~Barvinok的原始方法只能保证 $\|x\|_1 = O\bigl((n \cdot m)^n\bigr)$。为证明此结果，我们使用了新建立的组合Nullstellensatz的算法变体；2）对于任意 $p \in \bigl(R_{\geq 1} \cup \{\infty\}\bigr)$ 的范数 $\|\cdot\|_p$，该问题是NP难的。3）作为应用，我们证明：对于给定 $A \in Z^{m \times n}$ 和 $b \in Q^m$，计算多面体 $P = \{x \in R^n \colon A x \leq b\}$ 中整数点的问题，可通过一个具有 $O\bigl(ν^2 \cdot n^3 \cdot Δ^3 \bigr)$ 次操作的算法求解，其中 $ν$ 是 $P$ 的法扇形三角剖分的最大尺寸，$Δ$ 是 $A$ 的秩序子行列式的最大值。作为进一步应用，它为有界余维数多面体中的计数问题提供了更精细的复杂度界。例如，在无界子集和问题的多面体中。