The homeless exist because there is a market for them. They consume what would otherwise be waste. They give a second life to goods that are otherwise not valued. They recycle or consume things that would otherwise rot in a landfill. In this way, they are healthy part and a natural niche of any economy.
Therefore, you should not feel sorry for them. Do not give them money out of pity when you see them on the streets. Do not create labor for them out of pity. In doing so, you are reducing their efficiency and disturbing the natural population of homeless, thereby only contributing to the waste in your economy.
You should, however, treat them with respect. As self-employed waste management specialists, they are just as much a part of the economy as you are working for McDonald's or Microsoft. Instead, you should displace that disrespect to the government bureaucrats that provide "services" not essential to securing free production, no matter how nice their suits are, because they are the true freeloaders with no natural place in an economy. For a list of people to spit on in the streets and avert your eyes to, you can start here.
As long as there is a market for the homeless, they will continue to exist. They are scavengers, no more unnatural than vultures. The population of homeless is dependent on the availability of waste. Government sponsored "spread the work" schemes will not reduce the population of homeless because, even assuming no extra waste from these efforts, the resultant decline in purchasing power will only make homelessness more likely.
The way to decrease the homeless population is to decrease waste. Granted that 0 waste is a virtual impossibility, as it is a result of uncertainty in the market, it can be greatly reduced through the easing of government involvement in the market. For example, subsidizing the ailing American car industry only results in the production of goods that will not be consumed (waste). Furthermore, one cannot ignore the uncertainty imposed on the market by central banking. Therefore, the true way to end homelessness is by freely allowing the market to do what it does best: provide the highest quality goods and services to the most people in the most efficient manner possible.