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

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 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.