Abstract
If
is a collection of subsets ofS andw is a nonnegative weight function onS, the problem of selecting a subset belonging to
which has maximum weight is solved by a ‘greedy-type’ algorithm forall w if and only if
is the set of independent sets of a matroid. This result is then generalised to show that ‘greedy-type’ algorithms select an optimum forall linear functionsc·x; x in some compact set\(U \subseteq R^n \) andc > 0 if and only if the convex closure ofU is essentially a polymatroid. A byproduct of this is a new characterization of polymatroids.
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Dunstan, F.D.J., Welsh, D.J.A. A greedy algorithm for solving a certain class of linear programmes. Mathematical Programming 5, 338–353 (1973). https://doi.org/10.1007/BF01580137
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DOI: https://doi.org/10.1007/BF01580137