Four Nutriments and Dependent Origination
4 Nutriments is what is required to sustain life. Dependent Origination (DO) describes the same from the origination point of view.
DO does not mention Edible food which is one difference, Though it is needed to sustain the body.
Craving and clinging in DO relates to volitional thought as it is I or mine who subject of craving and clinging and these volitional thoughts.
Sense impression / contact is common to both. So is consciousness.
Though the 4 Nutriments do not cover feeling explicitly they are integral part of both:
"In that manner, I say, O monks, should the nutriment sense-impression be considered. If the nutriment sense-impression is comprehended, the three kinds of feeling are thereby comprehended. And if the three kinds of feeling are comprehended, there is, I say, no further work left to do for the noble disciple.
Similarly craving which is not explicitly a part of the 4 Nutrimens but found in the explanation hence an integral part of both:
In that manner, I say, O monks, should the nutriment volitional thought be considered. If the nutriment volitional thought is comprehended, the three kinds of craving are thereby comprehended. And if the three kinds of craving are comprehended, there is, I say, no further work left to do for the noble disciple.
SN 12.63 - Puttamansa Sutta
Samudaya Sutta presents this slightly differently to Puttamansa Sutta.
This perhaps can also be viewed as a condensation of DO.
Four Nutriments in the context of Satipathana
How this relates to Satipatthana is as follows:
a monk dwells exertive, clearly knowing, mindful, observing [ body | feelings | mind | dhammas ] in the [ body | feelings | mind | dhammas ], removing covetousness and displeasure in the world
(Magga) Vibhaṅga Sutta
"Removing covetousness and displeasure in the world" means you have to be equanimous to the sensations either pleasant or unpleasant originating from: body, feelings, mind, dhammas. (This also is found in Dhātu Vibhaṅga Sutta and Titth’ayatana Sutta rendering of the Satipatthana which makes it more clearer.) In either cases (pleasant or unpleasant [or neutral]) they are due to contact. To make sure your attention is wise attention you should contemplate on impermanence also. Pahāna Sutta also mentions how you should deal with sensation, i.e., to be equanimous than reacting with craving aversion and ignorance. Wise attention, knowing the 3 Characteristic (being free from Vipallasa - see Vipallasa Sutta), displaces ignorance. Cover coming craving or ensuring than any experience (pleasant, unpleasant or neutral) does not lead to craving ensures liberation:
And if the three kinds of craving are comprehended, there is, I say, no further work left to do for the noble disciple.
Cessation of Name-Form by starving yourself. It is by understanding
how consciousness arises due to fabrication. Karma / fabrication are formed from activities. This again can be overcome by not generating unwholesome roots, i.e., be equanimous to sensation seeing their impermanence.