WARSAW, Poland — Poland is so short of workers that even its employment offices are struggling to find staff.

A glut of vacancies has meant that lines for the permits required to hire foreign workers – a procedure that should last no more than two months – now extend for up to a year in parts of the country, according to consultancy Deloitte Poland.

Construction companies alone are in need of “thousands, maybe tens of thousands of foreign workers,” said Piotr Kledzik, chairman of the executive board at the Polish unit of Porr AG, an Austrian builder.

“It’s impossible for Poland to complete its investment projects without them,” Kledzik said. “And lines of Ukrainian applicants, instead of shrinking, extend forever.”

The arrival of an estimated 1.5 million Ukrainians in recent years has provided enough relief to keep Poland’s economy growing at its fastest pace in six years while taking pressure off wages. But strains are beginning to show. Years of emigration by Polish workers to the richer West, compounded by the government’s flat-footed response as the labor market tightens, sent domestic vacancies up nearly 40 percent last year.

Almost two-thirds of public-sector employers are seeking new personnel, hamstringing the government’s management of a labor shortage that risks becoming a major drag on the European Union’s biggest eastern economy. Poland’s employment offices have 380 vacancies for positions handling foreign job applications, Deputy Labor Minister Stanislaw Szwed said. That averages out to more than one per each outpost across the country.

“There are no people interested in jobs at public offices,” Szwed said at a conference. A computer system that’s expected to “fundamentally solve the issue” won’t be ready until 2019, he said.