A Compendium of Subset Search Problems and Reductions relating to the Parsimonious Property
Celina Janet Bartlett

TL;DR
This thesis explores the relationship between Subset Search Problems and Parsimonious reductions, providing a comprehensive analysis of their similarities, differences, and a collection of 46 reductions among classic NP-complete problems.
Contribution
It offers a detailed comparison of SSP and Parsimonious reductions, including a theorem delineating their properties, and compiles a large set of proven reductions for key computational problems.
Findings
SSP and Parsimonious reductions are similar but not equivalent.
A theorem characterizes conditions for reductions to preserve both properties.
Compiled 46 reductions among 30 classic NP-complete problems.
Abstract
This thesis centers around the concept of Subset Search Problems (SSP), a type of computational problem introduced by Gr\"une and Wulf to analyze the complexity of more intricate optimization problems. These problems are given an input set, a so-called universe, and their solution lies within their own universe, e.g. the shortest path between two point is a subset of all possible paths. Due to this, reductions upholding the SSP property require an injective embedding from the universe of the first problem into that of the second. This, however, appears inherently similar to the concept of a Parsimonious reduction, a reduction type requiring a bijective function between the solution spaces of the two problems. Parsimonious reductions are mainly used within the complexity class #P, as this class of problems concerns itself with the number of possible solutions in a given problem. These…
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Taxonomy
TopicsRough Sets and Fuzzy Logic · Logic, Reasoning, and Knowledge · Automated Road and Building Extraction
