No and your assumptions are incorrect
The pet overpopulation problem is DECREASING. The spay & neuter campaign has been very sucessful.
"In the 1970s, between 12 and 20 million cats and dogs were euthanized each year, according to the American Society for the Prevention of Cruelty to Animals (ASPCA) and the Humane Society International. This year, less than 4 million will be euthanized. Meanwhile, the number of pets that Americans own has increased. Stephen Zawistowski, science adviser for the ASPCA, told the AP that while in the '70's there were about 62 million pets, today there are over 170 million." http://theanimalrescuesite.greatergood.com/clickToGive/ars/article/Spaying-and-neutering-campaigns-reduce-euthanasia-rates521
There are additional issues of the need for responsibly bred pets, enforcement, etc.