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There were signs the air travel crisis was beginning to ease this week as lawmakers took up a bill to end the government shutdown, but it remained unclear when the federal government might lift its restrictions on air travel. The Trump administration and the airline industry have predicted it could take about a week after Congress passes legislation to reopen the government for air travel to return to what it was before the government shut down on Oct. 1. But Transportation Secretary Sean Duffy has promised to base his decisions on restoring flights at 40 key airports on data from the Federal Aviation Administration, meaning airlines may face another weekend of strain. “The weekends have been really, really tough,” Chris Sununu, who heads the trade association Airlines for America, said Wednesday at a news conference. “If the House acts and we see things moving forward, the hope is that F.A.A. or D.O.T. would be able to see their data points,” added Mr. Sununu, a former Republican governor of New Hampshire. “The big data point we look at is simply that staffing level. That’s the big one.” Officials pointed to a rise in absences among air traffic controllers, who have been forced to work without pay through the six-week shutdown, as a major factor in the department’s decision to restrict air travel. Thank you for your patience while we verify access. If you are in Reader mode please exit and log into your Times account, or subscribe for all of The Times. Thank you for your patience while we verify access. Already a subscriber? Log in. Want all of The Times? Subscribe.