Consider an edge-weighted graphG = (V, L), and define ak-cover C as a subset of the edgesL such that each vertex inV is incident to at least one edge ofC, and|C| = k. GivenG andk, the problem is to find ak-cover of minimum weight sum. This paper presents characterizations of minimumk-covers, and shows their weight to be convex with the parameterk. An efficient algorithm is presented which generates minimumk-covers continuously as the parameterk ranges over all feasible values, together with a proof of optimality. The computational order of this algorithm is found to be|V| |L| ².