"They committed acts of genocide against them."
"They commited the acts of genocide against them."
I think the second one is correct. We know what kind of acts they committed: "of genocide"
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For a plural noun, the choice of whether to include a definite article or to omit it depends on whether the speaker wishes to call out specific examples of the noun or if he simply wishes to speak of the noun in general without referencing a specific occurrence. To make the point easier, I'm going to substitute nouns into your examples (I just picked two old countries that don't exist anymore so don't read into it).
"Rome committed the acts of genocide against Gaul."
This statement implies that specific acts of genocide came up earlier in the conversation. Perhaps you were discussing a time when Rome burned down several towns in Gaul. In this case, it is correct (and required) to include "the" because you are trying to call out instances of "acts of genocide". To use the article this way, you should be able to substitute something earlier from the conversation into "the a acts of genocide" without changing the meaning.
"Rome committed acts of genocide against Gaul."
Rather than being a statement about specific instances of "acts of genocide", this is more of a general statement about "Rome", "Gaul", and "acts of genocide". The sentence with the definite article requires context to make sense (because the reader would not know what "the acts of genocide" refers to otherwise) whereas the sentence without the definite article can stand by itself. Said another way:
"Acts of genocide" has no greater meaning that its definition.
"The acts of genocide" refers to specific examples of the definition.
Briefly, in the singular, you do not have to option of omitting the article. You must choose between the and a/an. I suppose the theory behind this might be something like:
Articles are used to call out instances of nouns. Since you are speaking in the singular, you must be calling out one instance. So an article is required.
In the singular, "the" implies that the conversation mentioned one specific instance of the noun and "a/an" implies that the conversation did not mention a specific instance of the noun, but the speaker knows that it is singular.
The acts are not "of genocide". That makes no sense, because genocide is a type of act, not an act in itself. So one commits an act of genocide, such as putting Jews in a gas chamber.
In your examples, it really comes down to whether you're using a definite or indefinite article. The difference between them is the difference between talking about a specific act of genocide, or any old act of genocide whatsoever. The definite article is the word the, regardless of whether the noun it introduces is singular or plural.
Thus, if the specific acts of genocide were already previously mentioned, and you are referring back to them:
They committed the acts of genocide
But if you haven't explicitly specified which acts those were, then all we can say is:
They committed acts of genocide