Delta-modular ILP Problems of Bounded Co-dimension, Discrepancy, and Convolution (2405.17001v2)

Abstract: 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.