Bounds for Approximating Circuits

We have seen that some optimization problems cannot be solved by tropical circuits using a polynomial number of gates. But what happens if we allow the circuits to only output “almost optimal” values: can at least then the circuit size be substantially reduced? The goal of this chapter is to present recently emerged arguments for proving lower bounds on the size of approximating tropical circuits. It turns out that the task of proving lower bounds for approximating $$(\min ,+)$$ circuits is much easier than for $$(\max ,+)$$ circuits. While the size of the former circuits is lower bounded by the size of monotone Boolean circuits, approximating $$(\max ,+)$$ circuits are much more powerful. For them, neither the Boolean bound nor even counting arguments work. Still, also for $$(\max ,+)$$ circuits, the situation is not hopeless: there is a general combinatorial bound leading to strong lower bounds on the size of such circuits approximating explicit maximization problems, as well.