Algorithms for Standard-form ILP Problems via Koml\'os' Discrepancy Setting

TL;DR

This paper develops refined fixed-parameter tractable algorithms for standard-form integer linear programming problems using discrepancy theory, improving efficiency based on matrix discrepancy bounds.

Contribution

It introduces a novel approach combining discrepancy-based dynamic programming with matrix discrepancy bounds to solve ILPs efficiently parameterized by matrix properties.

Findings

Algorithms run in time polynomial in input size with exponential dependence on discrepancy bounds.

Using current bounds, the algorithms have quasi-polynomial dependence on k and polynomial on Δ.

Under the Komlós conjecture, the algorithms' dependence on k becomes exponential, simplifying analysis.

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 $\Delta$, the maximum absolute value of a $k\times k$ minor of $A$. Our approach combines discrepancy-based dynamic programming with matrix discrepancy bounds in Koml\'os' setting. Let $\kappa_k$ 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 $O(\kappa_k)^{2k}\Delta^2$, and the corresponding feasibility problem in time $O(\kappa_k)^k\Delta$. Using the best currently known bound $\kappa_k=\widetilde O(\log^{1/4}k)$, this yields running times $O(\log k)^{\frac{k}{2}(1+o(1))}\Delta^2$ and $O(\log k)^{\frac{k}{4}(1+o(1))}\Delta$, respectively. Under the Koml\'os conjecture, the dependence on $k$ in both running times reduces to $2^{O(k)}$.