Disjoint bases in a polymatroid

Let f : 2 → Z be a polymatroid (an integer-valued non-decreasing submodular set function with f(∅) = 0). We call S ⊆ N a base if f(S) = f(N). We consider the problem of finding a maximum number of disjoint bases; we denote by m be this base packing number. A simple upper bound on m is given by k = max{k : ∑ i∈N fA(i) ≥ kfA(N), ∀A ⊆ N} where fA(S) = f(A∪S)−f(A). This upper bound is a natural generalization of the bound for matroids where it is known that m = k. For polymatroids, we prove that m ≥ (1 − o(1))k/ ln f(N) and give a randomized polynomial time algorithm to find (1− o(1))k/ ln f(N) disjoint bases, assuming an oracle for f . We also derandomize the algorithm using minwise independent permutations and give a deterministic algorithm that finds (1− ǫ)k/ ln f(N) disjoint bases. The bound we obtain is almost tight since it is known there are polymatroids for which m ≤ (1 + o(1))k/ ln f(N). Moreover it is known that unless NP ⊆ DTIME(n log ), for any ǫ > 0, there is no polynomial time algorithm to obtain a (1 + ǫ)/ ln f(N)-approximation to m. Our result generalizes and unifies two results in the literature.