A Compendium of Reductions: reductions.network

TL;DR

reductions.network addresses the problem of organizing and navigating a large space of problem reductions by modeling problems as vertices and reductions as edges in graphs. The approach combines graph-based visualization with a searchable, filterable data repository and a three-layer software stack (Data Repository, Backend, Frontend) that supports community contributions and extension to new complexity classes. It catalogs networks for classical $NP$, $#P$, and $SSP-NP$, as well as parameterized $W[1]$, $W[2]$, and gap-preserving reductions tied to the PCP-Theorem and the Unique Games Conjecture, providing a practical tool to identify reduction pathways and clusters. The work offers an extendable, interactive resource that supports reproducibility through Markdown-based entries and CI, and facilitates education and collaboration via visual, navigable networks.

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.

Figures (5)

• Figure 1: The classical network.
• Figure 2: The Window for the problem Vertex Cover in the classical network.
• Figure 3: The window for the reduction from 3Satisfiability to Vertex Cover in the classical network.
• Figure 4: The navigation toolbar for the classical network.
• Figure 5: The layered architecture of the software.