Non-integral kernelization degrees for List H-Coloring-derived CSPs

Determine whether the restricted class of constraint satisfaction problems arising from List H-Coloring can exhibit tight non-integral kernelization degrees, as occurs for general constraint satisfaction problems.

Background

The paper relates List H-Coloring with a vertex-cover parameter to constraint satisfaction problems whose variables correspond to vertices in the vertex cover and whose constraints arise from vertices outside it. The invariants c^*(H) and d^*(H) describe, respectively, an arity-based upper bound and a polymorphism-based lower-bound parameter for these derived CSPs.

General CSPs are known to have problems with tight non-integral kernelization degrees, including examples with every rational exponent p/q greater than 1. It is unresolved whether this phenomenon can occur in the more restricted CSP class generated by List H-Coloring instances, which would bear directly on whether the conjectured integer exponent d^*(H) always gives the optimal kernel size.