Approximately covering by cycles in planar graphs

Let G = (V(G), E(G)) be a graph and let C be the collection of its cycles. Let p : E(G) --> Z(0)(+) be a non-negative, integer- valued function on its edge set. The CYCLE COVER PROBLEM is the optimization problem of finding a multiset alpha of cycles of G such that each edge e E E is in at least p(e) of the cycles in alpha and such that the sum of the lengths of all cycles in alpha is minimum. We will show how to approximate within a factor of 8 the optimum value of the cycle cover problem for planar graphs in polynomial time.