We study the problem of maximizing a monotone non-decreasing function f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} is submodular, the greedy algorithm will find a solution with value at least 12\documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{2}$$\end{document} of the optimal value under a general matroid constraint and at least 1-1e\documentclass[12pt]{minimal}
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\begin{document}$$1-\frac{1}{e}$$\end{document} of the optimal value under a uniform matroid (M=(X,I)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathcal {M} = (X,\mathcal {I})$$\end{document}, I={S⊆X:|S|≤k}\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {I} = \{ S \subseteq X: |S| \le k\}$$\end{document}) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least 11+μ\documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{1+\mu }$$\end{document} of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where μ=α\documentclass[12pt]{minimal}
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\begin{document}$$\mu = \alpha $$\end{document}, if 0≤α≤1\documentclass[12pt]{minimal}
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\begin{document}$$0 \le \alpha \le 1$$\end{document}; μ=αK(1-αK)K(1-α)\documentclass[12pt]{minimal}
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\begin{document}$$\mu = \frac{\alpha ^K(1-\alpha ^K)}{K(1-\alpha )}$$\end{document} if α>1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha > 1$$\end{document}; here α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is a constant representing the “elemental curvature” of f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document}, and K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document} is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a 1-(α+⋯+αk-11+α+⋯+αk-1)k\documentclass[12pt]{minimal}
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\begin{document}$$1 - (\frac{\alpha + \cdots + \alpha ^{k-1}}{1+\alpha + \cdots + \alpha ^{k-1}})^k$$\end{document} approximation under a uniform matroid constraint. Under this unified α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-classification, submodular functions arise as the special case 0≤α≤1\documentclass[12pt]{minimal}
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\begin{document}$$0 \le \alpha \le 1$$\end{document}.
