Integer knapsack problems with profit functions of the same value range

Integer knapsack problems with profit functions of the same value range are studied. Linear time algorithms are presented for the case of convex non-decreasing profit functions, and an NP-hardness proof and a fully polynomial-time approximation scheme are provided for the case of arbitrary non-negative non-decreasing profit functions. Fast solution procedures are also devised for the bottleneck counterparts of these problems. Computational complexity of the case with concave profit functions remains open.