We study the standard-form ILP problem $\max\{ c^\top x \colon A x = b,\; x \in Z_{\geq 0}^n \}$, where $A\in Z^{k\times n}$ has full row rank. We obtain refined FPT algorithms parameterized by $k$ and $Δ$, the maximum absolute value of a $k\times k$ minor of $A$. Our approach combines discrepancy-based dynamic programming with matrix discrepancy bounds in Komlós' setting. Let $κ_k$ denote the maximum discrepancy over all matrices with $k$ columns whose columns have Euclidean norm at most $1$. Up to polynomial factors in the input size, the optimization problem can be solved in time $O(κ_k)^{2k}Δ^2$, and the corresponding feasibility problem in time $O(κ_k)^kΔ$. Using the best currently known bound $κ_k=\widetilde O(\log^{1/4}k)$, this yields running times $O(\log k)^{\frac{k}{2}(1+o(1))}Δ^2$ and $O(\log k)^{\frac{k}{4}(1+o(1))}Δ$, respectively. Under the Komlós conjecture, the dependence on $k$ in both running times reduces to $2^{O(k)}$.

翻译：我们研究标准形式的ILP问题 $\max\{ c^\top x \colon A x = b,\; x \in Z_{\geq 0}^n \}$，其中 $A\in Z^{k\times n}$ 具有满行秩。我们以 $k$ 和 $Δ$（$A$ 的 $k\times k$ 子式绝对值的最大值）为参数，获得了精细化的FPT算法。我们的方法将基于差异的动态规划与Komlós设定中的矩阵差异界相结合。设 $κ_k$ 表示所有具有 $k$ 列且列欧几里得范数至多为 $1$ 的矩阵的最大差异。在输入规模的多项式因子范围内，优化问题可在 $O(κ_k)^{2k}Δ^2$ 时间内求解，相应的可行性问题可在 $O(κ_k)^kΔ$ 时间内求解。利用当前已知的最优界 $κ_k=\widetilde O(\log^{1/4}k)$，这分别产生运行时间 $O(\log k)^{\frac{k}{2}(1+o(1))}Δ^2$ 和 $O(\log k)^{\frac{k}{4}(1+o(1))}Δ$。在Komlós猜想成立的条件下，两个运行时间中对 $k$ 的依赖均降至 $2^{O(k)}$。