Intractability of approximate multi-dimensional nonlinear optimization on independence systems

We consider optimization of nonlinear objective functions that balance d linear criteria over n -element independence systems presented by linear-optimization oracles. For d = 1 , we have previously shown that an r -best approximate solution can be found in polynomial time. Here, using an extended Erdős–Ko–Rado theorem of Frankl, we show that for d = 2 , finding a ρ n -best solution requires exponential time.