It seems to me that the root of the problem in pet overpopulation can be expressed simply in economic terms. The supply of pets is way, way greater than the demand for them. What we need to do is reach an equilibrium where the supply for pets equals the demand.
Many groups and individuals are working very hard on this by trying to decrease the supply. The primary tool for this is spaying and neutering to decrease the population. A second strategy is limiting the number of breeders out there who simply keep adding to the supply for their own personal profit.
Some may argue that we should also increase the demand, but I think the demand for pets, such as it is, is actually artificially high. Many people don't want animals...they want toys. They want something they can take home for their kids or something they can use as an accessory for a while. Then, when the work sets in, they dump the animals....pumping them back into the supply chain.