Hyperplanes Avoiding Problem and Integer Points Counting in Polyhedra

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. More precisely, it appears when one needs to evaluate certain rational generating functions in an avoidable critical point. 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 \(\|x\|_1 = O\bigl((n \cdot m)^n\bigr)\); 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 \(O\bigl(\nu^2 \cdot n^3 \cdot \Delta^3 \bigr)\) operations, where \(\nu\) is the maximum size of a normal fan triangulation of \(P\), and \(\Delta\) is the maximum value of rank-order subdeterminants of \(A\). It refines the previous state-of-the-art \(O\bigl(\nu^2 \cdot n^4 \cdot \Delta^3\bigr)\)-time algorithm.