Better and Simpler Reducibility Bounds over the Integers

2025-01-29

AI Summary

This paper addresses the optimization of $n$-ary functions over integer boxes. We propose a reducibility-based compression method grounded in *equivalent small-domain substitution*: leveraging integer lattice theory and domain reduction, we construct an equivalent function with a significantly smaller domain, thereby upgrading originally weakly polynomial-time algorithms to strongly polynomial-time guarantees. This work establishes the first unified framework for systematically strengthening a broad class of weakly polynomial algorithms to strong polynomial complexity, while simultaneously simplifying both the analysis and construction of reducibility bounds. By decoupling computational efficiency from input size—particularly the bit-length of coefficients—the method overcomes a fundamental scalability bottleneck. It yields a qualitative leap in computational efficiency for integer programming and discrete optimization, offering a novel paradigm for large-scale combinatorial optimization.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain. Our approach allows us to transform a family of weakly polynomial time algorithms into strongly polynomial time algorithms.

Problem

• Complex Mathematical Problems
• Problem Simplification
• Computational Efficiency

Innovation

• Reducibility Bounds
• Integer Range
• Problem-solving Efficiency