Algorithms for Standard-form ILP Problems via Komlós' Discrepancy Setting

Abstract: 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 $O(κ_k)<sup>{2k}Δ<sup>2$, and the corresponding feasibility problem in time $O(κ_k)<sup>kΔ$. Using the best currently known bound $κ_k=\widetilde O(\log<sup>{1/4}k)$, this yields running times $O(\log k)<sup>{\frac{k}{2}(1+o(1))}Δ<sup>2$ and $O(\log k)<sup>{\frac{k}{4}(1+o(1))}Δ$, respectively. Under the Komlós conjecture, the dependence on $k$ in both running times reduces to $2<sup>{O(k)}$.