What cobra is suggesting is somewhat different than filing a criminal complaint, he's suggesting a civil lawsuit, not a criminal one, for failure to do their job. I think that would be a writ (or order) of mandamus. Here's what Wikipedia has to say about those:
A writ of mandamus or simply mandamus, which means "we order" in Latin, is the name of one of the prerogative writs in the common law, and is a court order directing someone, most frequently a government official, to perform a specified act. Traditionally, the act to be compelled must be one that is "ministerial" rather than "discretionary," which means it must not involve any qualitative judgment to tell whether it has been done (or done right or completely): Signing a document or handing one over to someone is ministerial; providing some service is discretionary, whether it is painting a portrait or removing a gall bladder or cutting hair or typing a letter. (In that sense, "ministerial" has a "binary" meaning—the act is either done or not done).
Mandamus in the United States
In general
In the administrative law context in the United States, the requirement that mandamus can be used only to compel a ministerial act has largely been abandoned. By statute or by judicial expansion of the writ of mandamus in most of the U.S. states, acts of administrative agencies are now subject to judicial review for abuse of discretion. Judicial review of agencies of the United States federal government for abuse of discretion is authorized by the Administrative Procedure Act.
Here's their link to the Administrative Procedures Act
I am exploring criminal complaints. Not civil suits. Thanks for the link.