After riding the elevator to the top floor of a building in the Louis H. Pink Houses in Brooklyn and scanning the rooftop, the police officers made their way back down, floor by floor, searching the stairwells and hallways. On the sixth floor, slumped against the stairs, was a man who said he was waiting for his ex-girlfriend. The officers ran a warrant check on the 49-year-old man and arrested him after learning he was wanted for a parole violation.

“You never know what you’re going to find,” said one of the officers, Sgt. Marshall Winston, who has policed public housing for 23 years. “Sometimes we catch somebody with a gun. Other times we catch somebody with drugs. And sometimes you just catch somebody down on their luck.”

The patrols, known as verticals, are painstaking police work, and for the New York City Housing Authority, they do not come cheap. About 2,000 officers are assigned to the projects, and Nycha, as the authority is known, pays the Police Department about $70 million a year. The payment is a legacy of the mergers that brought the transit and housing authority police forces into the New York Police Department almost 20 years ago.

But the housing authority’s increasingly strained finances have focused attention on the payments, and Mayor-elect Bill de Blasio, who will be sworn in on Wednesday, has promised to end them. At a forum during the mayoral campaign, Mr. de Blasio said the money “was taken on the assumption that Nycha was just awash in federal money, all these wonderful resources coming into Nycha. And that hasn’t been true for decades.”