They spend vast amounts of money on the problem, but the way they approach the problem makes the solution intractable. To the extent the drug addicted and severely mentally ill homeless can want things, and want the right to continue their current life, they are afforded that right. That is to say, it is their fundamental right to use drugs, live, and defecate on the street (I know these things are illegal, but really, legality is determined by enforcement, not what is on the books. And as far as enforcement goes, these things are legal for all intents and purposes).

So long as they don't want help, and so long as it is their right to refuse and continue as they are, nothing will change. The alternatives are many, but they all start with some type of enforcement agency willingly stripping these peoples rights, and forcibly and potentially violently taking them into government custody, which the citizens of San Francisco find far more appalling than solving the problem.

Note: What I said above doesn't generalize to 'the homeless,' and people out of a job and down on their luck, It specifically is targeted to the worst offenders who scatter the city with waste and needles.