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      "statement": "# Greedy selection: a guarantee when benefits have diminishing returns\n\nNemhauser, Wolsey and Fisher established a sharp worst-case guarantee for greedily maximizing a nondecreasing submodular set function under a cardinality limit.\n\nSuppose you can choose only a few locations, services or resources. A greedy method adds the option that gives the greatest remaining benefit. This is simple, but usually does not find the exact optimum. The 1978 paper identified a reusable guarantee when benefit never decreases and has diminishing returns: an option becomes no more valuable when it is added to a larger existing selection. For a budget of K selections and zero benefit at the empty set, greedy achieves at least 1-(1-1/K)^K of the best possible value, approaching about63 percent for large K. The paper also supplied worst-case examples and analyzed separate local-swap, grouped-greedy, relaxation and partial-enumeration approaches. The guarantee concerns the stated benefit model, exact marginal comparisons and a limit on the number of selected elements. It does not cover arbitrary interactions, nonmonotone benefits or every resource constraint. Earlier facility-location results and matroid/submodularity theory remain distinct contributions.\n\n## Scientific scope\n\nThe greedy approximation uses the stated finite-set, monotone submodular objective and cardinality constraints. Diminishing returns, feasibility and normalization assumptions matter. Nonmonotone functions and arbitrary constraints do not inherit the same guarantee. An approximation ratio is not exact optimization or measured practical performance.\n\n## People and systems\n\nG. L. Nemhauser; L. A. Wolsey; M. L. Fisher\n\n\n\n## Original sources\n\n- [Nemhauser, Wolsey and Fisher, An analysis of approximations for maximizing submodular set functions-I,1978originaljournal CMUcoursecopy](https://www.cs.cmu.edu/~aarti/Class/10704_Spring15/submod-max.pdf)\n- [Springer publisherCrossrefdeposit for1978original](https://api.crossref.org/works/10.1007%2FBF01588971)\n",
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