They loved their new apartment in Southeast Washington. It felt like a new beginning, far from the homeless shelter at the former D.C. General Hospital, with its rats and cold showers and the methadone clinic crowd hanging out at the bus stop.

“But we just couldn’t keep up,” said the mother, a 50-year-old woman with chronic health issues, an 8-year-old daughter and a bed back at the shelter.

People are suddenly paying attention to the shelter because of the disappearance of 8-year-old Relisha Rudd, who lived there for nearly two years before she vanished with a janitor suspected of killing his wife and then shooting himself.

It’s a depressing place, and even worse the second time around, the woman and her daughter said. The two were supposed to be one of the success stories of the city’s Rapid Re-Housing Program, which is the centerpiece of Mayor Vincent C. Gray’s campaign to house “500 families in 100 days.”

Rapid Re-Housing takes the $50,000 it costs each year to keep a family at the shelter and uses it to subsidize the first few months of rent for an apartment. The generous subsidy — often about 60 percent — can be renewed twice for fourth months at a time, theoretically giving a family time to get back on its feet.

It sounds like a good deal and a good way to help the skyrocketing number of homeless families in the nation’s capital. City officials say that between 60 and 80 percent of families who enter the Rapid Re-Housing Program do not end up back at the shelter.

But I have my doubts about those numbers, and so do folks such as Kate Coventry, a policy analyst at the D.C. Fiscal Policy Institute, who said the success of the program is difficult to measure. The city hasn’t been tracking what happens to families once their subsidies run out unless they reenter the shelter.

As for the homeless themselves, the word at the shelter when it comes to Rapid Re-Housing is: “Don’t do it.”

“You wind up right back here. Who wants that?” a woman standing outside of D.C. General told me Friday afternoon, just as the crowd from the methadone clinic next door to the shelter was getting rowdy and a man tried to sell us stolen gallons of Tide detergent. “For me to stay in just about any apartment here, I’d have to make $40,000 a year. How is that going to happen in four months?”

Contessa Allen-Starks has been a poster child on government Web sites for the program. Last summer, she was featured in The Washington Post as one of the parents who was working with Rapid Re-Housing.

She’s a bright, hardworking mom who fell on tough times and landed at the shelter. Even while she was living there, she kept the job she’s had for seven years, working part time as a Giant pharmacy tech just across the state line in Maryland.

She got the subsidy and an apartment, which wasn’t far from work and was a safe, comfortable place for her to figure out her next move: whether to work full time at the pharmacy or try to turn an unpaid internship at a doctor’s office into a full-time job.

A half a year later?

“I’m not doing as well as I’d hoped,” said Allen-Starks on a busy day at the pharmacy. “I lost the apartment, and I’m living with a friend now.”

After a couple of extensions of the subsidy, the government withdrew its help paying her $979 monthly rent. She and her two children didn’t land back at the homeless shelter.

“I’m never going back there,” she declared. But statistically, because she never returned to the place that scarred her and her children, she is counted by the city as one of the 60 to 80 percent who benefited from Rapid Re-Housing.

The program might work if the city’s housing market were more affordable. Unfortunately, rents and housing prices are continuing to soar.

“I think that affordable housing is really key,” said Jenny Reed, policy director at the D.C. Fiscal Policy Institute, who has been studying the city’s wildly changing housing market for years.

“Rapid Re-Housing is a tool to end homelessness quickly. It gets you into a safer place more quickly. But it is not an affordable-housing tool,” she said.

Even if a family manages to stay in an apartment after the subsidy runs out, some are paying more than half their incomes in rent. And that’s a scary, unsustainable situation, Reed said.

We’ve got more than 1,000 residents coming into the city every month, median rents have risen more than half again as fast as inflation, and the city’s affordable housing stock has been cut in half.

The deck is stacked against the city’s homeless families. Maybe Gray will be able to get 500 families out of the shelter and the motels where they’ve been living. But if more isn’t done to increase the city’s stock of affordable housing, many of those families will be right back in the shelter next winter.

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