Sell or Hold: A simple two-stage stochastic combinatorial optimization problem

There are n individual assets and k of them are to be sold over two stages. The first-stage prices are known and the second-stage prices have a known distribution. The sell or hold problem (SHP) is to determine which assets are to be sold at each stage to maximize the total expected revenue. We show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We also identify a polynomially solvable case of SHP when the number of scenarios in the second stage is constant. A max{ 1 2 , k n }-approximation algorithm is presented for the scenario-based SHP. An example shows that the bound is tight.