It was a statement born of confidence and boldness. Mayor Michael R. Bloomberg declared in 2004 that he would do what no mayor of his era had done: reduce the city’s homeless population by two-thirds by the time he left office.

The mayor called upon the same traits he is now highlighting, as some say he is paving the way for a presidential bid: private sector know-how, non-ideological approaches, cutting-edge technology and a willingness to take on entrenched issues.

But despite a number of initiatives, including computer tracking and prevention programs, the population of homeless families, after dipping in 2005, reached its highest point in two decades in May.

Families complain that the worst problems of the early 1980s are back, citing overcrowding, late-night busing to overnight beds and filthy conditions. The city disagrees with these characterizations.

Some leading advocates for the homeless, who had taken a hiatus after more than 20 years of litigation against the city, are going back to court, saying that the families the city is turning away are sleeping on the subways and boardwalks — something the city also disputes.

The disappointing results from what was perhaps the mayor’s most ambitious poverty initiative underscore the limits of Mr. Bloomberg’s style of governing, critics say.

His administration spent $79 million for new tools and programs to solve the problem. A new computer system tracked which neighborhoods homeless people came from and how long they stayed in the system, mimicking the Compstat system that the police used to fight crime.

The mayor reached out to some of the most implacable critics of the city’s homeless policy, inviting them in on press conferences and planning sessions.

Then, most controversially, he took aim at what his administration believed was a perverse incentive for those living marginally to become homeless — a city policy that gave homeless families top priority for federal housing vouchers. The city replaced generous federal grants with time-limited rental assistance meant to help out in an emergency while discouraging dependency.

Mr. Bloomberg said he had made the system both “more humane and more accountable.” He and his deputy mayor for health and human services, Linda I. Gibbs, said the goal of a two-thirds reduction in homelessness was more of an inspirational goal than a concrete target.

Still, in a written statement, the mayor acknowledged that the rising number of homeless families was “not acceptable.”

Although Mr. Bloomberg wins almost universal praise for tackling a complex issue, the change in awarding federal housing vouchers did not bring about a sustained reduction in requests for shelter. And at the same time, critics say that the city’s rental assistance plan was insufficient and that it permitted families to rent only the most decrepit apartments.

While the computer system helped pinpoint the neighborhoods where homeless people were most likely to come from, prevention programs offering one-time rent assistance and help with arrears in rent have not been as successful as expected. Neighborhoods with such programs have seen slower increases in the number of homeless families, but increases nonetheless.

Mr. Bloomberg’s conservative critics say that he was naïve in expecting it to be productive to bring advocates for the homeless into policy and planning discussions. The advocates, the critics say, are always going to be pushing for more housing and more spending on the homeless, regardless of what the city does.

“What we are seeing are the limits of the technocratic approach,” said Edmund J. McMahon, a senior policy fellow with the Manhattan Institute for Policy Research. “No one questions the competency or thoughtfulness of Bloomberg on this issue. It is more the overall strategy of thinking you can work the problem through and thinking everyone will be reasonable.”

There is no question that Mr. Bloomberg inherited an overwhelmed system. Families routinely slept overnight on benches of the notorious intake office in the Bronx. A legal battle between the city and advocates left the most minute details of operation, like the availability of milk bottle warmers, up to the courts.

The mayor assigned the task of straightening out the system to Ms. Gibbs, as commissioner of the Department of Homeless Services. Her early innovations and energy captured Mr. Bloomberg’s imagination and he set some big targets. With much fanfare, he set the goal of a two-thirds reduction in five years.

Ms. Gibbs, who has since been promoted to deputy mayor, acknowledged some setbacks, but bristled at criticisms of the administration’s record.

She pointed out that the number of single homeless adults had dropped 19 percent, to about 6,700, since 2004, and is at its lowest in seven years. Families now make up about 80 percent of the homeless population, and individuals 20 percent.

She said that there had also been improvements for families. Eligibility screening times have been cut in half and no one sleeps in intake offices overnight anymore.

And she predicted that the number of homeless families would decline after more policy changes take effect. For example, the city increased the monthly rental allowance to make it more competitive in the New York market, and those who receive assistance are no longer penalized for earning outside income.

“It is still just a matter of adjusting the policy until it is right,” she said.

But Steven Banks, attorney-in-chief of the Legal Aid Society, which represents the plaintiffs in the 24 years of litigation, argued that the Bloomberg strategy needed more than a little tweaking.

“Many of those policies had significant flaws, because there is an underlying sense that if you take away shelter or permanent housing resources from homeless people, somehow they will go away,” he said. “The facts on the ground of the last year have shown this is a mistake.”

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