Homeland Security has built up a backlog of more than 1.2 million illegal immigrants who it believes have overstayed visas but managed to arrest only about 3,400 of them, according to the most recent data, which works out to a rate of about 1 in every 350 lawbreakers.

That is far worse than the rate for those who crossed the border illegally, and it means criminals, people engaged in narcoterrorism and other national security risks are left to run free in the U.S., the Homeland Security inspector general said in a report Thursday.

Federal agents have trouble tracking down the criminals because the government still doesn’t monitor departures, meaning it can’t be certain whether those who came on tourist, business or student visas leave when they are supposed to.

Officers have to check as many as 27 in-house systems, in addition to state databases, to try to guess whether someone has left. Even then, they can make catastrophic mistakes when it wrongly appears that a visitor has left the country.

“Such false departure information resulted in [deportation] officers closing visa overstay investigations of dangerous individuals, such as suspected criminals, who were actually still in the United States and could pose a threat to national security,” the investigators said in the report. “For example, an ERO officer stated that a suspect under investigation was listed as having left the country, but had given his ticket to a family member and was still residing in the United States.”

Visa overstays, as they are called, have become an increasing focus of the immigration debate. As the flow of illegal border crossings declines, an increasing percentage of those in the country illegally are travelers who came on business, tourist or student visas but didn’t leave when their time was up.

Several of the Sept. 11, 2001, hijackers were overstays.

Homeland Security has struggled to get a handle on the situation, or even to figure out how bad it is.

A report last year looking at just a portion of visas calculated that more than 500,000 visitors overstayed in 2015. The total backlog grew to more than 1.2 million, the inspector general said.

Meanwhile, U.S. Immigration and Customs Enforcement officers arrested only 3,402 potential overstays in 2015.

In its official response, ICE said it is trying to do a better job of calculating the number of visa overstays.

President Trump has pushed the Homeland Security Department to finish the system that would track departures, and tests are being run at Hartsfield-Jackson International Airport in Atlanta.

The department also plans to release its next overstay report soon. That report will cover almost all visa categories, so the government will have a better sense for how bad the problem is.

But Rep. Bob Goodlatte, chairman of the House Judiciary Committee, said that until the government tracks all departures, it won’t know what’s going on.

“Visa security is a matter of national security, and it is imperative that we know who is coming to our country and when they leave so that we protect American citizens and our interests,” the Virginia Republican said.

Congress demanded a biometric entry-exit system more than 20 years ago, but administrations in both parties have failed to deliver, saying that airports aren’t configured to check departures and that the land ports of entry are an even bigger logistical hurdle.

Mr. Goodlatte said he expects Mr. Trump’s focus on immigration to finally push Homeland Security to finish the job.

Until then, officers will waste time on bogus leads, the inspector general’s report said. The data are so unreliable that officers and agents often end up finding an overstay still in the country who the systems said already had left, or spend time trying to track down someone who did leave the country or obtained legal status.

“An ICE officer estimated that he spent more than 50 hours on a single suspect, only to find the individual had applied for [an immigration] benefit and should not have been categorized as an overstay,” the audit said.

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