Delta-modular ILP Problems of Bounded Codimension, Discrepancy, and Convolution (new version)

Abstract

For integers $k,n \geq 0$ and a cost vector $c \in Z^n$, we study two fundamental integer linear programming (ILP) problems: \[ \text{(Standard Form)} \quad \max\bigl\{c^\top x \colon Ax = b,\ x \in Z^n_{\geq 0}\bigr\} \text{ with } A \in Z^{k \times n}, \text{rank}(A) = k, b \in Z^k, \] \[ \text{(Canonical Form)} \quad \max\bigl\{c^\top x \colon Ax \leq b,\ x \in Z^n\bigr\} \text{ with } A \in Z^{(n+k) \times n}, \text{rank}(A) = n, b \in Z^{n+k}. \] We present improved algorithms for both problems and their feasibility versions, parameterized by $k$ and $\Delta$, where $\Delta$ denotes the maximum absolute value of $\text{rank}(A) \times \text{rank}(A)$ subdeterminants of $A$. Our main complexity results, stated in terms of required arithmetic operations, are: \[ \text{Optimization:}\quad O(\log k)^{2k} \cdot \Delta^2 / 2^{\Omega(\sqrt{\log \Delta})} + 2^{O(k)} \cdot \text{poly}(\varphi), \] \[ \text{Feasibility:} \quad O(\log k)^k \cdot \Delta \cdot (\log \Delta)^3 + 2^{O(k)} \cdot \text{poly}(\varphi), \] where $\varphi$ represents the input size measured by the bit-encoding length of $(A,b,c)$. We also examine several special cases when $k \in \{0,1\}$, which have important applications in: expected computational complexity of ILP with varying right-hand side $b$, ILP problems with generic constraint matrices, ILP problems on simplices. Our results yield improved complexity bounds for these specific scenarios. As independent contributions, we present: An $n^2/2^{\Omega(\sqrt{\log n})}$-time algorithm for the tropical convolution problem on sequences indexed by elements of a finite Abelian group of order $n$; A complete and self-contained error analysis of the generalized DFT over Abelian groups in the Word-RAM model.