In New York State, billions of dollars have flowed through this patchwork of programs in recent years, although precise figures are hard to compile.

The projects under scrutiny in the two investigations, which prosecutors are conducting with the transportation authority’s inspector general, are not the only major undertakings by Schiavone and Skanska in the city. In joint ventures with a third company, they are also involved in some of the authority’s most ambitious efforts. They have a $1.14 billion contract to build the tunnels and station structures for the No. 7 subway line extension and a $337 million contract to build some of the tunnels for the Second Avenue subway, though no accusations of wrongdoing have been raised in connection with those projects.

The schemes at the center of the two investigations are simple: Rather than hire a minority- or women-owned subcontractor or a company the federal government has certified as a struggling business, the contractors ran the payroll for their own workers  or paid another subcontractor  through a minority company that served as a “pass through.”

In the case of Schiavone, based in Secaucus, N.J., part of the scheme centered on a company hired to haul away dirt excavated from the water-treatment plant project and work at the $530 million South Ferry subway station renovation and at the Times Square station. The company did a tiny part of the work, while most of it was performed by another company, a mob-connected trucking firm, the people briefed on the case said.

Skanska, those with knowledge of that case said, paid workers for dewatering and demolition at the Fulton Street Transit Center site through a company called Environmental Energy Associates in Ridgefield, N.J. That company, a certified disadvantaged business enterprise, or D.B.E., collected a fee for the service it provided, handling just the payroll, the people said.

Representatives of the offices of the United States attorney in Brooklyn, Loretta E. Lynch, and the Manhattan United States attorney, Preet Bharara, declined to comment on Tuesday.

Several law enforcement officials who investigate construction corruption acknowledged that the goals for the participation of minority- and women-owned contractors and disadvantaged businesses often did not accurately reflect the universe of qualified companies.