A Compendium of Reductions: reductions.network

TL;DR

reductions.network is an interactive online database visualizing problems and their reductions across various complexity classes, facilitating exploration, search, filtering, and community contributions to advance complexity theory research.

Contribution

it introduces a comprehensive, graph-based platform for exploring and contributing to reductions among problems in multiple complexity classes, integrating diverse reduction types.

Findings

Includes classical, parameterized, and gap-preserving reduction networks.

Enables visualization of problem clusters and reduction pathways.

Supports community contributions and expansion to new complexity classes.

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.