Hyperplanes Avoiding Problem and Integer Points Counting in Polyhedra
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2024
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| author | Dakhno, Grigorii Gribanov, Dmitry Kasianov, Nikita Kats, Anastasiia Kupavskii, Andrey Kuz'min, Nikita Moiseev, Stanislav |
| author_facet | Dakhno, Grigorii Gribanov, Dmitry Kasianov, Nikita Kats, Anastasiia Kupavskii, Andrey Kuz'min, Nikita Moiseev, Stanislav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07030 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperplanes Avoiding Problem and Integer Points Counting in Polyhedra Dakhno, Grigorii Gribanov, Dmitry Kasianov, Nikita Kats, Anastasiia Kupavskii, Andrey Kuz'min, Nikita Moiseev, Stanislav Computational Complexity Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics 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. |
| title | Hyperplanes Avoiding Problem and Integer Points Counting in Polyhedra |
| topic | Computational Complexity Computational Geometry Discrete Mathematics Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2411.07030 |