Most definitely. I don't particularly agree that it should be painless or quick either. Personally I say let the family of the victim have a go at them, but that's just me.
On the topic of it being cheaper to house an inmate, in the US a single prison meal can range from .15-.40. So if we go off of that: .15 * 356 = $53.4 a year if they're fed once a day. The "recommendation" amount of meals for a prisoner is 3 meals a day, so let's do: $53.4 * 3 = $160.2 a year. That's just for food. Now, there's already been studies on this, and it costs an average of more than $31,000 per inmate, per year, according to the Vera Institute of Justice.
We try to take care of prisoners even when they're dying for some odd fucking reason. The Death Penalty is upwards of a few hundred thousand to a million, but in actuality it should be somewhere around $100 at most. They're dying for their crimes, a case of bullets isn't that much, neither is a gun. No one wants death on their hands anymore.