The problem is especially striking for innovation-based companies, which are generating jobs at a rapid clip as technology spreads through every sector of the economy. By 2015, about 900,000 information and communications technology vacancies may go unfilled in the European Union, the European Commission warned in a recent report on the digital economy. The gap “is of major concern to European competitiveness” and to the economy as a whole, the commission said.

Governments and companies around Europe are fast-tracking efforts to retrain the unemployed for a burst of technology-related jobs. They are also stepping up campaigns to lure university students to mathematics, engineering and science in place of popular courses in the humanities and social sciences.

In Ireland, the government introduced a series of retraining and higher-education programs and sought to polish the allure of mathematics degrees as alarm bells sounded over the issue a couple of years ago. At the time, unemployment was around 14 percent after an economic collapse that destroyed jobs in the construction sector, which had employed around a quarter of the young men in the country.

Multinational technology and social media companies kept investing, lured by Ireland’s ultralow 12.5 percent corporate tax rate and an English-speaking work force. But many have been forced to look outside the country for employees with the right skills, even with 282,900 unemployed in Ireland and a jobless rate of around 12.5 percent.

The issue peaked last summer, when PayPal’s chief executive in Ireland, Louise Phelan, stoked controversy by acknowledging that the company had recruited from 19 other countries for 500 positions in its operations center in Dundalk because of a lack of foreign-language skills among Irish nationals. This summer, Fujitsu, which employs 800 people in Ireland, revealed that it had had to hire most of its Ph.D.-level experts from abroad.

All told, around half of information technology jobs in Dublin were being filled with foreign workers, while around 4,500 information technology jobs in the country were going unfilled because of a limited supply of suitably skilled applicants, various studies have shown. Paul Sweetman, the director of ICT Ireland, a business lobby group, said that part of Ireland’s strategy was to enhance its attractiveness as an investment and work destination by luring bright minds from around the world to the technology sector.

The skills shortage prevented Ireland-based companies from “effectively executing their business strategies,” which created a risk of lower productivity and slower growth, according to a recent report by the consulting company Accenture.