Euthanizing stray cats is not only morally wrong, but it doesn't hinder the stray cat population either. It has been proven not to work, many, many times.

Removing cats from an area is a futile effort; one that is bound to fail. Municipalities engaged in any type of catch and kill efforts are fighting a cruel, endless, losing battle against nature that is a waste of taxpayer dollars. When one group of cats is removed, another set moves in to the newly vacated space, and breed at an astounding rate.  This is called a vacuum effect.  The cat population is never really reduced, just rearranged. The only answer to solving cat over-population is T-N-R (trap, neuter, and return).  Not only does it decrease the population over time, but it improves the cats' lives as well.  By neutering male cats, their urine will not be as strong, and they will not feel the need to spray.  For spaying female cats, they will not yowl while looking for a mate, and there will be less fighting among cats. One study found a 66% decrease in population over 11 years, and another 10-year study found a 32% decrease in colony size, by performing T-N-R. 

The idea of "don't feed them, they'll go away," is not exactly correct.  The cats are still there, hunting the animals that live nearby.  Feral cats don't need cat food to survive, they live off of nature.  However, they will eat the cat food placed out for them, as it is almost like a treat (just like people and candy; you can't live off of it, but it doesn't mean you don't have some when it's around). 

In addition, when there is "no mouse problem," and cats are attempting to be removed, there will be a spike in rodent populations (resulting from cats not hunting them).  That'll add another problem on top of the cat over-population, because catching and killing simply does not work.  

I am an animal lover, and I believe it is incredibly cruel and ridiculous to kill any animal, especially cats, simply because they live outdoors and are considered a "bother." Cats have lived outdoors for thousands of years, and only relatively recently became domesticated.  To kill them in their natural element is morally wrong, and it doesn't solve any any of the problems that are being faced.  Try an idea that has been proven to work, for both humans and cats alike; T-N-R!