Washington ? Americans have yet to find a Republican they’d clearly prefer over President Barack Obama, although half say the president does not deserve re-election.

Among Republicans, the desire to oust Obama is clear, according to a new AP-GfK poll. But it has not resolved divisions over the choice of a nominee. Former Massachusetts Gov. Mitt Romney is reasonably popular, but he has not pulled away from the field.

Former pizza company executive Herman Cain runs close to Romney as the candidate Republicans would most like to see on the ballot, but many Republicans are reluctant to back a man who has never held office. Texas Gov. Rick Perry lags in the poll, which was conducted before Tuesday night’s combative debate in Las Vegas.

In that two-hour forum, several candidates sharply criticized Cain’s tax proposals, and a newly energized Perry hit Romney hard on immigration.

In the poll, Romney was the choice of 30 percent of Republicans, with Cain about even at 26 percent. Perry was preferred by 13 percent, and Rep. Ron Paul of Texas topped the list of those in single digits.

Among all adults surveyed, half said Obama should not be re-elected, and 46 percent said he should be. That continues his gradual slide since May.

When all adults are asked about hypothetical head-to-head matchups, Obama and Romney run almost even, 48 percent for Obama to 45 percent. Obama holds a narrow edge over Cain, 49 percent to 43 percent. He leads Perry, 51 percent to 42 percent.

Luis Calderon of El Monte, Calif., exemplifies those unhappy with Obama but not ready to dump him.

“Even though I criticize him, I still want him to win,” said Calderon, 56, a self-employed handyman who was laid off by an oil company three years ago. Obama “has to get down to business, forget about promises, just do it, create jobs,” Calderon said. “But in order to create jobs, he has to be harder on the Republicans.”

A Democrat, Calderon said Romney “is the one that may do a little dent on Obama.”

Romney spent four years as Massachusetts governor, and he ran for president in 2008. Cain is the only candidate who has never held elected office, which might present some problems. Americans have no recent history of electing inexperienced politicians as president except war hero Dwight Eisenhower.

Of the Republicans polled, about four in 10 say they’re less inclined to vote for someone who has never been elected to public office. That’s far more than say they are disinclined to vote for a Mormon, a woman or a black candidate.

Romney and former Utah Gov. Jon Huntsman are Mormons. Rep. Michele Bachmann of Minnesota is the only woman in the race. Cain is black.

Nineteen percent of Republicans, and 21 percent of all adults, say they are less likely to vote for someone who is a Mormon. Anne Fish, a Republican and retired teacher from Columbus, Ohio, is among them. Fish, 73, said she would not support Romney “because he is not a Christian.”

Mainstream Mormons, including Romney, consider themselves Christians.

Fish said she probably will support Perry. “Although I have some doubts, I think he has some ideas about how to improve the economy, how to help our country develop more jobs,” she said.

Ronald Wilson, a conservative Republican from Bucyrus, Ohio, said he’s undecided, although “I favor Herman Cain. He’s not infected by Washingtonitis.”

Wilson, 65, a retired stone quarry worker, called Romney “better than nothing.”

Such comments underscore Romney’s challenge. Many GOP insiders see him as the most plausible nominee and Obama’s strongest potential challenger. But Romney generates little passion among Republican voters, who seem to keep shopping for an alternative as time ticks down to the Jan. 3 Iowa caucuses.