Reviews: Counting the Optimal Solutions in Graphical Models

The authors deal with a new problem that extends existing problems in a natural way, and present new solutions that extend existing solutions in a natural way; thus in some ways the paper is not very original, but it must be noted that the problem #opt has some surprising properties that make the whole endeavor quite interesting. As for the contribution, it is a valuable piece with new algorithms that have good performance. I really take issue with the emphasis on semirings; I cannot see how this helps the whole effort. Are semirings bringing new insights here? Or are they just more general so that the algorithm applies to other problems?