Plowing with precedence in polynomial time

The plowing with precedence problem is a variant of the windy postman problem, where a plow is required to clean streets after a heavy snowfall with traversing costs depending on the direction of traversal as well as whether a street has been previously plowed or not. We prove that this problem can be solved in polynomial time under natural cost structures. We also propose heuristics for this problem, which compare favorably with the state of the art.