NEW YORK (AP) – A less expensive Sept. 11 memorial that cuts back on underground space and its museum won tentative state approval Friday, while families said they couldn’t support a design that doesn’t change the way the names of their loved ones are listed.

The Lower Manhattan Development Corp. moved forward with a developer’s proposal to cut soaring costs for the memorial complex, which includes twin reflecting pools with waterfalls marking the destroyed World Trade Center, a plaza with oak trees and an underground museum.

The agency’s board agreed to proceed with environmental and historic reviews of the new memorial plans.

Gov. George Pataki and Mayor Michael Bloomberg ordered a redesign of the “Reflecting Absence” memorial after contractors estimated in May that it could cost close to $1 billion to build.

The developer, Frank Sciame, said this version would cost $510 million, although an additional $100 million or so in costs to make the site suitable for building were taken out of the budget and transferred to the government agency that owns the trade center site.

The new design moves the names of the nearly 3,000 people killed in the 2001 terrorist attacks from parapets surrounding the below-ground pools to street level, a major victory for family members who said it would dishonor their loved ones if they had to go underground to mourn them.

The memorial’s architect, Michael Arad, had sought to keep the names underground, saying mourners could have a peaceful space below ground to view names surrounding the pools.

Arad also sought to list the names of those killed at the trade center, at the Pentagon and in Shanksville, Pa., randomly around the pools, saying it would reflect the chaos of their deaths.

Family members and the city’s police and fire unions have said for two years that they wanted their loved ones grouped with the people they worked with or died with.

“If they are listed inappropriately, the fact that they moved them above ground isn’t meaningful,” said Edith Lutnick, whose brother was killed in the attack.

Debra Burlingame, whose brother was the pilot on the hijacked plane that crashed at the Pentagon, noted that the change wouldn’t add to the memorial’s cost. “What sense does it make to separate a family of four who died on an airplane?” she asked.

LMDC President Stefan Pryor declined to comment Friday on the issue of how the names would be listed. Officials have said they would defer to Arad’s proposal, which would also identify rescue workers with insignia.

On his weekly radio show, Bloomberg said it wouldn’t be appropriate to list names as the families have requested.

“It would be so complex, and it’s not clear _ do you want to be next to your sibling or with your fellow workers, because it may have been in different places,” the mayor said.

He said he favored recognizing fire, police and emergency workers, “but I think that they also should be mixed in with all of the other names because they were human beings, we’re all human beings.”

The LMDC has committed $250 million to build the memorial, while a private foundation that suspended fundraising in May has raised $131 million. The foundation was poised to launch a national advertising campaign next week.

Preliminary construction on the memorial began in March, then stopped two months later while the design was reconsidered. Officials said construction would resume soon, but couldn’t say when. Officials hope to open the memorial in three years.