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

TL;DR

This paper introduces improved algorithms for bounded codimension ILP problems, providing tighter complexity bounds and exploring special cases with applications in computational complexity and algebraic transforms.

Contribution

It presents new algorithms with enhanced complexity bounds for ILP problems parameterized by codimension and discrepancy, and analyzes the tropical convolution and generalized DFT over Abelian groups.

Findings

Optimized algorithms with subexponential complexity in terms of elta

Special case algorithms for k=0,1 with improved bounds

Error analysis of generalized DFT in Word-RAM model

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.