Homeless Problem is one of the many Side Quests in South Park: The Stick of Truth.
To start this quest, go and talk to the Mayor in City Hall and she will tell you that she needed help getting rid of the homeless people that are all over town.
Note that they are really tough to fight so before attempting the battle make sure you are levelled up a bit.
In total there are seven homeless “camps”. To eradicate a camp of homeless people, just head there and fight the homeless people.
You can start with the easiest one, the loner homeless guy right outside Kenny’s house. head there and take him out.
U-Stor-It storage facility
There is one homeless found in the back of the U-Stor-It storage facility inside the truck the one parked right outside the building of the same name. There are boxes behind the truck, use them to climb inside, to fight the homeless.
Note that this homeless have got a rat with him.
By the bus Stop
In a clearing near the bus stop you will find a trio of homeless men near your own home. Head to the left of the bus stop (you will see the girl on the bench) and there is a large tree with a bird blocking a path before get to Stark’s Pond.
Go behind the tree to reveal the hidden path through the broken fence. Follow the path and you will find a wooden barricade, smash it and you will find the homeless guys.
Beat all three guys until the leave. It is best to attempt this once you are at least level 3.
Now the rest of homeless people are found in the sewers you cant miss them. There are a total of four camps in the sewers. But before starting exploring the sewers you will need the alien anal probe teleportation device. Complete the main quest Alien Abduction to get this.
After acquired the device now you get through the gap in the sewers and fight the homeless people. Once inside the sewer make your way into Mr. Hankey house and you can pick up his side quest Dropping The Kids Off.
Continue onward and climb up the ladder and then head to the right to find the first homeless camp (one homeless).
Proceed a little further to the right and go down the second ladder, you will find another homeless camp on the left.
Head back to by Mr. Hankey's house. Go all the way right down here, then shoot a ladder to bring it down. Then walk to the left, and shoot a knob to stop the flow of sewage to create the path.
Go down the path and then to the right you will find another homeless person.
Go all the way to the right until you see wood pallet leaned against the wall. Smash it to expose a lever. Pull it, and then go down the ladder it opens and you will see the breakable wall to the right, instead go to the left to find the last homeless camp.
Note: The hobos in the sewer often come in pairs and usually have rats with them and sometimes stronger mutant variety.
Once you have cleared the sewers go back to Mayor McDaniels at the City Hall to tell her that you have solved the town’s problem with homeless people. You will received the Mayor's freindship.