Perhaps I feel too strongly about this, but I think that certain people should absolutely be executed, but probably in rare cases. Vince Weiguang Li comes to mind.

I understand that the murderer was mentally ill, but I refuse to believe that he didn't understand what he was doing. He got onto a bus, pulled out a butcher knife, and beheaded an innocent person and ate part of the victim's body. He took the knife with him on the bus, and sat next to his victim for no apparent reason, so clearly the murder was premeditated. He knew he was going to kill someone when he got on that bus, and he knew he was going to kill the victim when he sat by him. He brandished the victim's head to the other passengers and tried to escape afterwards, so he knew that he had done something despicable. People who do something like that can never be trusted in the general population again, so you either lock them up forever or you execute them. It makes no sense to keep them locked up for life, so I'd say it only makes sense to execute them.

I have no problem with executing somebody like this because I don't think a life has sacred value simply because they are a homo sapiens. For example, if you had a dog that was not malnourished, but still decided to rip somebody apart, then that dog would be put down. It may be in that dog's nature, but it would be too dangerous to have near people after that and it would be cruel to just abandon the dog somewhere, so the only option is to put it down. The same goes for a human. Just because you're a member of our species, your life isn't sacred. Even if we pretend that human life in general is some sacred gem, there's still 7 billion of us to go around. Getting rid of a handful of despicable people will not now, and will not ever be missed in the universe.