The discrete sell or hold problem with constraints on asset values

The discrete sell or hold problem (DSHP), which is introduced in He et al. [6], is studied under the constraint that each asset can only take a constant number of different values. We show that if each asset can only take two different values, the problem becomes polynomial-time solvable. However, even if each asset can only take three different values, the DSHP is still NP-hard. An approximation algorithm, which could potentially achieve a better approximation ratio than the one proposed in [6], is also given under this setting.