Rural postman parameterized by the number of components of required edges

In the Directed Rural Postman Problem (DRPP), given a strongly connected directed multigraph D=(V,A) with nonnegative integral weights on the arcs, a subset R of required arcs and a nonnegative integer ℓ, decide whether D has a closed directed walk containing every arc of R and of weight at most ℓ. Let k be the number of weakly connected components in the subgraph of D induced by R. Sorge et al. [30] asked whether the DRPP is fixed-parameter tractable (FPT) when parameterized by k, i.e., whether there is an algorithm of running time O⁎(f(k)) where f is a function of k only and the O⁎ notation suppresses polynomial factors. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time O⁎(2k) when ℓ is bounded by a polynomial in the number of vertices in D. The same result holds for the undirected version of DRPP.