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

2026-04-10

AI Summary

This work addresses the standard form of integer linear programming—maximizing \( c^\top x \) subject to \( Ax = b \), \( x \in \mathbb{Z}_+^n \), with \( A \) full row rank—and presents the first systematic application of Komlós discrepancy theory to this setting. By integrating matrix discrepancy bounds with discrepancy-driven dynamic programming, the authors devise a refined fixed-parameter algorithm parameterized by the number of constraints \( k \) and the maximum absolute value \( \Delta \) of any \( k \times k \) subdeterminant of \( A \). In the general case, the optimization problem is solvable in \( O((\log k)^{k/2(1+o(1))} \Delta^2) \) time, while feasibility can be decided in \( O((\log k)^{k/4(1+o(1))} \Delta) \) time. Assuming the Komlós conjecture holds, the dependence on \( k \) improves dramatically to \( 2^{O(k)} \), substantially advancing the state-of-the-art complexity bounds.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $O(κ_k)^{2k}Δ^2$, 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 $2^{O(k)}$.

Problem

• Integer Linear Programming
• Standard-form ILP
• Discrepancy Theory
• Komlós Conjecture
• Fixed-parameter Tractability

Innovation

• integer linear programming
• discrepancy theory
• fixed-parameter tractability
• Komlós conjecture
• dynamic programming