Can algorithm approximations be combined in some form for purposes of multi-objective optimization? The study of approximation algorithms is very new to me, but I have been having a lot of difficulty searching for more information on this, and the textbook I skimmed through didn't seem to cover multi-objective problems (unless a different name is used).
Let's say I have two objectives, so we can formulate the problem as:
$$ \underset{x}{\arg\max} ~~a(x) + \lambda~b(x) $$
And I want to solve the problem using a greedy approach. My understanding is that this is a submodularity-preserving operation, so the $(1-1/\epsilon)$ error bound would hold, but only if $a(x)$ and $b(x)$ are both submodular.
However, what if $a(x)$ or $b(x)$ is not submodular?
Consider, for example, the case where $a(x)$ is a vector of values, so I just need the index of the maximum value (submodular, easy problem), and solving function $b(x)$ alone using a greedy algorithm is 2-approximate. Can I say anything about the accuracy of the combined problem?