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 two concepts, SSP and Parsimonious reductions, are inherently similar but, crucially, not equivalent. We therefore explore the interplay between reductions upholding the SSP and Parsimonious properties, highlighting both the similarities and differences by providing a comprehensive theorem delineating the properties required for reductions to uphold both attributes. We also compile and evaluate 46 reductions between 30 subset search variants of computational problems, including those of classic NP-complete problems such as Satisfiability, Vertex Cover, Hamiltonian Cycle, the Traveling Salesman Problem and Subset Sum, providing reduction proofs, illustrative examples and insights as to where the SSP and Parsimonious properties coexist or diverge. With this compendium we contribute to the understanding of the computational complexity of bilevel and robust optimization problems, by contributing a vast collection of proven SSP- and #P-complete problems.

翻译：本论文围绕子集搜索问题（SSP）这一概念展开，该计算问题由Gr\"une与Wulf提出，用于分析更复杂优化问题的计算复杂性。此类问题的输入为一个称为“全域”的集合，其解则位于该全域内部，例如两点间最短路径即是所有可能路径的子集。基于此特性，保持SSP性质的归约需要从第一个问题的全域到第二个问题全域的单射嵌入。然而，这本质上与简约归约的概念高度相似——后者要求两个问题的解空间之间存在双射函数。简约归约主要应用于复杂度类#P中，因为此类问题关注给定问题可能解的数量。SSP与简约归约这两个概念本质相似，但关键之处在于二者并不等价。因此，我们通过构建一个系统定理来阐明同时满足两种属性所需的归约条件，从而深入探究保持SSP性质与简约性质的归约之间的相互作用，并着重比较其异同。此外，我们系统整理并评估了30个计算问题的46种子集搜索变体之间的归约关系，涵盖经典NP完全问题如可满足性问题、顶点覆盖问题、哈密顿回路问题、旅行商问题及子集和问题，同时提供归约证明、示例解析，并阐明SSP性质与简约性质在何种情况下共存或分离。通过本纲要，我们贡献了大量经证明的SSP完全问题与#P完全问题案例，为深入理解双层优化与鲁棒优化问题的计算复杂性提供了重要参考。