Solution #1: Identification, safe confinement.
The critical yet simple key to this aspect of the homeless animal problem is some form of identification so that a lost animal can get back home. I.D. tags are the most basic form of identification, are widely available, and come in all shapes and sizes. Microchip identification is becoming more and more common, and provides a permanent form of identification that can't be separated from the animal. All dogs and cats - even those who are indoor-only - should always wear some form of identification.
In addition, safely confining companion animals, not allowing them to roam, always knowing where they are and looking for them the minute it is realized they are missing, are simple yet crucial practices that would dramatically reduce the numbers of lost and stray animals entering animal shelters each year.