Polynomial-time kernel reductions

Today, the computational complexity of equivalence problems such as the graph isomorphism problem and the Boolean formula equivalence problem remain only partially understood. One of the most important tools for determining the (relative) difficulty of a computational problem is the many-one reduction, which provides a way to encode an instance of one problem into an instance of another. In equivalence problems, the goal is to determine if a pair of strings is related, so a many-one reduction with access to the entire pair may be too powerful. A recently introduced type of reduction, the kernel reduction, defined only on equivalence problems, allows the transformation of each string in the pair independently. Understanding the limitations of the kernel reduction as compared with the many-one reduction improves our understanding of the limitations of computers in solving problems of equivalence. We investigate not only these limitations, but also whether classes of equivalence problems have complete problems under kernel reductions. This paper provides a detailed collection of results about kernel reductions. After exploring possible definitions of complexity classes of equivalence relations, we prove that polynomial time kernel reductions are strictly less powerful than polynomial time many-one reductions. We also provide sufficient conditions for complete problems under kernel reductions, show that completeness under kernel reductions can sometimes imply completeness under many-one reductions, and finally prove that equivalence problems of intermediate difficulty can exist under the right conditions. Though kernel reductions share some basic properties with many-one reductions, ultimately the number and size of equivalence classes can prevent the existence of a kernel reduction, regardless of the complexity of the equivalence problem. The most important open problem we leave unsolved is proving the unconditional existence of a complete problem under kernel reductions for some basic complexity classes that are well-known to have complete problems under many-one reductions. Copyright 2010, 2011, 2012, 2014, 2015 Jeffrey Finkelstein 〈[email protected]〉.