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.

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