Efficiently Explaining CSPs with Unsatisfiable Subset Optimization

Recently, a novel method for explaining solutions of constraint satisfaction problems was proposed.
An explanation in that method is a sequence of simple inference steps, where the simplicity of an inference step depends on the number and types of constraints used, eventually explaining all logical consequences of the problem.
The current paper tackles two questions left in builds on these formal foundations and tackles two trailing questions: namely how to generate explanations that are provably optimal (with respect to the given cost metric) and how to generate them efficiently.
To answer these questions, we develop 1) an implicit hitting set algorithm for finding optimal unsatisfiable subsets; 2) a method to reduce multiple calls for (optimal) unsatisfiable subsets to a single call that takes constraints on the subset into account, and 3) a method for re-using relevant information over multiple calls to these algorithms.