For $k, n \geq 0$, and $c \in Z^n$, we consider ILP problems \begin{gather*} \max\bigl\{ c^\top x \colon A x = b,\, x \in Z^n_{\geq 0} \bigr\}\text{ with $A \in Z^{k \times n}$, $rank(A) = k$, $b \in Z^{k}$ and} \max\bigl\{ c^\top x \colon A x \leq b,\, x \in Z^n \bigr\} \text{ with $A \in Z^{(n+k) \times n}$, $rank(A) = n$, $b \in Z^{n+k}$.} \end{gather*} The first problem is called an \emph{ILP problem in the standard form of the codimension $k$}, and the second problem is called an \emph{ILP problem in the canonical form with $n+k$ constraints.} We show that, for any sufficiently large $\Delta$, both problems can be solved with $$ 2^{O(k)} \cdot (f_{k,d} \cdot \Delta)^2 / 2^{\Omega\bigl(\sqrt{\log(f_{k,d} \cdot \Delta)}\bigr)} $$ operations, where $ f_{k,d} = \min \Bigl\{ k^{k/2}, \bigl(\log k \cdot \log (d + k)\bigr)^{k/2} \Bigr\} $, $d$ is the dimension of a corresponding polyhedron and $\Delta$ is the maximum absolute value of $rank(A) \times rank(A)$ sub-determinants of $A$. As our second main result, we show that the feasibility variants of both problems can be solved with $$ 2^{O(k)} \cdot f_{k,d} \cdot \Delta \cdot \log^3(f_{k,d} \cdot \Delta) $$ operations. The constant $f_{k,d}$ can be replaced by other constant $g_{k,\Delta} = \bigl(\log k \cdot \log (k \Delta)\bigr)^{k/2}$ that depends only on $k$ and $\Delta$. Additionally, we consider different partial cases with $k=0$ and $k=1$, which have interesting applications. As a result of independent interest, we propose an $n^2/2^{\Omega\bigl(\sqrt{\log n}\bigr)}$-time algorithm for the tropical convolution problem on sequences, indexed by elements of a finite Abelian group of the order $n$. Additionally, we give a complete, self-contained error analysis of the generalized Discrete Fourier Transform for Abelian groups with respect to the Word-RAM computational model.

翻译：对于 $k, n \geq 0$ 和 $c \in Z^n$，我们考虑以下整数线性规划问题：\begin{gather*} \max\bigl\{ c^\top x \colon A x = b,\, x \in Z^n_{\geq 0} \bigr\}\text{ 其中 $A \in Z^{k \times n}$, $rank(A) = k$, $b \in Z^{k}$，以及} \max\bigl\{ c^\top x \colon A x \leq b,\, x \in Z^n \bigr\} \text{ 其中 $A \in Z^{(n+k) \times n}$, $rank(A) = n$, $b \in Z^{n+k}$。} \end{gather*} 第一个问题称为\emph{余维 $k$ 标准形式下的整数线性规划问题}，第二个问题称为\emph{具有 $n+k$ 个约束的典范形式下的整数线性规划问题}。我们证明，对于任意足够大的 $\Delta$，这两个问题均可在 $$ 2^{O(k)} \cdot (f_{k,d} \cdot \Delta)^2 / 2^{\Omega\bigl(\sqrt{\log(f_{k,d} \cdot \Delta)}\bigr)} $$ 次运算内求解，其中 $ f_{k,d} = \min \Bigl\{ k^{k/2}, \bigl(\log k \cdot \log (d + k)\bigr)^{k/2} \Bigr\} $，$d$ 为对应多面体的维数，$\Delta$ 是 $A$ 的 $rank(A) \times rank(A)$ 阶子式的最大绝对值。作为我们的第二个主要结果，我们证明这两个问题的可行性变体可在 $$ 2^{O(k)} \cdot f_{k,d} \cdot \Delta \cdot \log^3(f_{k,d} \cdot \Delta) $$ 次运算内求解。常数 $f_{k,d}$ 可替换为仅依赖于 $k$ 和 $\Delta$ 的其他常数 $g_{k,\Delta} = \bigl(\log k \cdot \log (k \Delta)\bigr)^{k/2}$。此外，我们考虑了 $k=0$ 和 $k=1$ 的不同特例，这些特例具有有趣的应用。作为一项具有独立意义的结果，我们针对由阶为 $n$ 的有限阿贝尔群元素索引的序列，提出了一个时间复杂度为 $n^2/2^{\Omega\bigl(\sqrt{\log n}\bigr)}$ 的热带卷积问题算法。此外，我们针对 Word-RAM 计算模型，对阿贝尔群上的广义离散傅里叶变换给出了完整且自洽的误差分析。