Let f:2^N --> \cZ^+ be a polymatroid (an integer-valued non-decreasing submodular set function with f(emptyset) = 0). A k-polymatroid satisfies that f(e) <= k for all e in N. We call a subset S of N independent if f(S) equals the sum of f(e) over the elements e of S and f(e) > 0 for all e in S. Finding a maximum-size independent set in a 2-polymatroid has been studied and polynomial-time algorithms are known for linear polymatroids. For k >= 3, the problem is NP-hard, and an approximation algorithm with ratio approaching 2/k is known and is obtained by swapping as long as possible a "large" subset from the current solution by a set with one more element. Here we give a simple analysis of the more particular two-for-one repeated swapping heuristic, obtaining a (weaker) 2/(k+1)-approximation.

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