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This paper considers the multiset selection problem with size constraints, which arises in many real-world applications such as budget allocation. Previous studies required the objective function f to be submodular, while we relax this assumption by introducing the notion of the submodularity ratios (denoted by α_f and β_f). We propose an anytime randomized iterative approach POMS, which maximizes the given objective f and minimizes the multiset size simultaneously. We prove that POMS using a reasonable time achieves an approximation guarantee of max{1-1/e^(β_f), (α_f/2)(1-1/e^(α_f))}. Particularly, when f is submdoular, this bound is at least as good as that of the previous greedy-style algorithms. In addition, we give lower bounds on the submodularity ratio for the objectives of budget allocation. Experimental results on budget allocation as well as a more complex application, namely, generalized influence maximization, exhibit the superior performance of the proposed approach.