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 Erdos-Ko-Rado theorem of Frankl, we show that for d = 2, finding a -best solution requires exponential time.