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We consider optimization of nonlinear objective functions that balance $d$linear criteria over $n$-element independence systems presented bylinear-optimization oracles. For $d=1$, we have previously shown that an$r$-best approximate solution can be found in polynomial time. Here, using anextended Erd\H{o}s-Ko-Rado theorem of Frankl, we show that for $d=2$, finding a$\rho n$-best solution requires exponential time.

Intractability of approximate multi-dimensional nonlinear optimization on independence systems