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 ρn-best solution requires exponential time. © 2011 Elsevier B.V. All rights reserved.