Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More

2026-01-02

arXiv.org

AI Summary

This work addresses the long-standing challenge of achieving super-polynomial speedups for weighted NP-hard problems—such as the Traveling Salesman Problem (TSP), weighted Max-Cut, and edge-weighted k-Clique—whereas their unweighted counterparts have seen significant algorithmic advances. The authors introduce the doubling constant of the weight set as a key parameter and leverage a constructive Freiman theorem to compress arbitrary weights into polynomially bounded integers. By combining dynamic programming over (min, +) or (max, +) semirings with polynomial embedding techniques, they effectively reduce weighted instances to near-unweighted complexity regimes. When the weight set exhibits a small doubling constant, this approach yields substantially improved running times, breaking free from traditional pseudo-polynomial dependencies and establishing an efficient bridge between weighted and unweighted problem formulations.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups. Currently, the only way to repurpose the algorithm of the unweighted version for the weighted version is to employ a polynomial embedding of the input weights. This, however, introduces a pseudo-polynomial factor into the running time, which becomes impractical for arbitrarily weighted instances. In this paper, we introduce a new way to repurpose the algorithm of the unweighted problem. Specifically, we show that the time complexity of several well-known NP-hard problems operating over the $(\min, +)$ and $(\max, +)$ semirings, such as TSP, Weighted Max-Cut, and Edge-Weighted $k$-Clique, is proportional to that of their unweighted versions when the set of input weights has small doubling. We achieve this by a meta-algorithm that converts the input weights into polynomially bounded integers using the recent constructive Freiman's theorem by Randolph and W\k{e}grzycki [ESA 2024] before applying the polynomial embedding.

Problem

• weighted problems
• doubling constant
• TSP
• Max-Cut
• pseudo-polynomial time

Innovation

• doubling constant
• weighted problems
• Freiman's theorem
• polynomial embedding
• NP-hard optimization