A Compendium of Reductions: reductions.network

2025-11-06

AI Summary

Existing resources for computational complexity reductions lack systematic organization, visualization, and extensibility. Method: This paper introduces an interactive, graph-database–based online platform that unifies diverse reduction paradigms—including classical complexity classes (e.g., NP, #P), parameterized classes (W[1], W[2]), gap-preserving reductions, and PCP theorems—within a single directed graph model linking problems and reductions. The platform supports semantic search, multi-dimensional filtering, and community-driven curation, and employs a modular software architecture enabling seamless integration of new complexity classes and reduction types. Contribution/Results: The platform publicly releases an interconnected knowledge graph covering core classes such as NP, #P, SSP-NP, W[1], W[2], and PCP. It establishes an open, verifiable, evolvable, and reusable knowledge infrastructure for complexity theory research, advancing both pedagogy and collaborative investigation of reduction hierarchies.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The website reductions.network serves as a comprehensive database for exploring problems and reductions between them. It presents several complexity classes in the form of an interconnected graph where problems are represented as vertices, while edges represent reductions between them. This graphical perspective allows for identifying problem clusters and simplifying finding problem candidates to reduce from. Moreover, users can easily search for existing problems via a dedicated search bar, and various filters allow them to focus on specific subgraphs of interest. The design of the website enables users to contribute by adding new problems and reductions to the database. Furthermore, the software architecture allows for the integration of additional graphs corresponding to new complexity classes. In the current state, the following networks with their respective complexity classes are included: - classical complexity including the classes NP, #P, and SSP-NP - parameterized complexity including the classes W[1], W[2] - gap-preserving reductions under the PCP-Theorem and the Unique Games Conjecture.

Problem

• Creating a comprehensive database for exploring computational problems and their reductions
• Visualizing complexity classes as interconnected graphs with problems and reductions
• Enabling users to search, filter, and contribute to complexity theory networks

Innovation

• Graph-based visualization of complexity classes
• Interactive search and filtering for problem discovery
• Extensible architecture supporting multiple reduction types