Final answer:
The main criterion for a protein to be processed and presented via the class I MHC pathway in an APC is that it is encoded by a viral gene.
Step-by-step explanation:
The main criterion that determines whether a protein is processed and presented via the class I MHC pathway in an antigen-presenting cell (APC) is that the protein is encoded by a viral gene. Intracellular antigens, such as those produced by viruses, are processed in the cytosol by the proteasome and then brought into the endoplasmic reticulum, where they interact with class I MHC molecules. These antigens are eventually transported to the cell surface for presentation to T cells.