In Sections 5.1 of The Design of Approximation Algorithms by Williamson and Shmoys, they describe a basic randomized algorithm for MAX SAT and how to derandomize it. The algorithm is just to assign each variable 1 (true) with probability 1/2 and 0 (false) with probability 1/2. In other words, sample uniformly at random from the space of all solutions. They show that this is a 1/2-approximation.
Then in Section 5.2, they describe how to derandomize it using the method of conditional expectations. (I won’t describe the process here because it is not very complex and widely known I’m assuming.)
My question is, why bother derandomizing this way? Or even, why bother making the algorithm random in the first place?
It seems to me that an equally good algorithm would be the one-liner which deterministically sets all variables to 1. Given some MAX SAT instance as input, it seems to me that you would also expect this to (i.e., "in expectation it would") satisfy half of the clauses. To me, the analysis of the random algorithm really seems to say that any fixed guess is "good." (Rather than showing that our random algorithm is inherently good.) So why go through the process of randomizing and derandomizing in the first place?
Thanks in advance!