Exploring the Reductions Between SSP-NP-complete Problems and Developing a Compendium Website Displaying the Results

TL;DR

This paper explores SSP reductions, compiles SSP-NP-complete problems, discovers new reductions, and develops an extendable website to visualize problem relations, aiding researchers in understanding complex problem interconnections.

Contribution

It introduces 19 new SSP reductions, proves 8 new SSP-NP-completeness results, and creates a comprehensive, extendable web-based compendium for SSP problems and reductions.

Findings

19 new SSP reductions identified

8 new SSP-NP-completeness proofs established

Developed an extendable compendium website for SSP problems

Abstract

SSP reductions are a type of polynomial reductions that also preserve the solutions of the instances. This means there is a mapping from each solution in the original instance to one in the reduced instance, allowing direct deduction of an original solution from a solution in the reduced instance. SSP reductions can be used to show SSP-NP completeness of a problem, which is interesting because it has been proven that two min-max variants of SSP-NP complete problems are $\Sigma_2^p$-complete. Min-max optimization problems are optimization problems with the objective of minimizing the maximum outcome. An example is a power network prone to attack by terrorists. Lets say the government wants to minimize the damage that can be achieved by the terrorists, e.g. by protecting the electricity poles which were found to be most important for distributing power. Most theoretical computer scientists assume min-max optimization problems to generally be $\Sigma_2^p$-complete, and with the help of the SSP-framework, this was shown for network interdiction and min-max regret robust optimization. This paper is devoted to the exploration of SSP reductions and the collection of SSP-NP-complete problems as well as SSP reductions. Additionally, a compendium website is developed, in which the SSP-NP-complete problems and SSP reductions are displayed in a graph, so that it becomes easy to understand the relations between the problems. The website is also extendable to include problems from other complexity classes and reductions that are not SSP reductions, such as gap-preserving reductions, fixed parameter tractable reductions or fine-grained reductions. Overall, 19 new SSP reductions are found, resulting in eight new SSP-NP-completeness proofs, and the compendium website is developed to enable theoretical computer scientists to easily access the results and provide a knowledge base for future research.