Why did Odysseus decide not to kill the cyclops in The Odyssey?
I do not disagree with the first answer -- Odysseus does say the lines mentioned in that answer. However, I think that there are a couple other reasons he did not kill the Cyclops.
First of all, his best chance to do so would have been in the cave while the Cyclops slept. If he had done so, how would he have moved the rock that blocked the opening to the cave? They would have been stuck.
You might also say that he wanted the Cyclops to be able to tell everyone of Odysseus's deeds. So he might have left the Cyclops alive to add to his own fame.
According to the ending of Book IX, Odysseus was fearful that, in fact, he considered it to be too difficult to kill the cyclops. He states, "I wish I could be sure of killing you outright and send you down to the house of Hades, as I am that it will take more than Poseidon to cure that eye of yours."
Also, consider the fact that he didn't want to provoke Poseidon (or Zeus) even further, which would only delay his voyage home.