Parameterized complexity of k-Chinese Postman Problem

We consider the following problem called the k-Chinese Postman Problem (k-CPP): given a connected edge-weighted graph G and integers p and k, decide whether there are at least k closed walks such that every edge of G is contained in at least one of them and the total weight of the edges in the walks is at most p? The problem k-CPP is NP-complete, and van Bevern et al. (to appear) and Sorge (2013) asked whether the k-CPP is fixed-parameter tractable when parameterized by k. We prove that the k-CPP is indeed fixed-parameter tractable. In fact, we prove a stronger result: the problem admits a kernel with O(k log k) vertices. We prove that the directed version of k-CPP is NP-complete and ask whether the directed version is fixed-parameter tractable when parameterized by k.