Intractability of approximate multi-dimensional nonlinear optimization on independence systems

TL;DR

This paper investigates the computational complexity of approximating multi-dimensional nonlinear optimization problems over independence systems, revealing exponential difficulty for certain cases when the number of criteria increases.

Contribution

It extends previous work by proving that for two criteria, finding near-optimal solutions is exponentially hard, using an advanced combinatorial theorem.

Findings

Finding a ρn-best solution for d=2 requires exponential time.

Polynomial-time solutions exist for d=1.

Complexity increases with the number of criteria.

Abstract

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 Erd\H{o}s-Ko-Rado theorem of Frankl, we show that for $d=2$, finding a $\rho n$-best solution requires exponential time.