Resources for Nedarim 82
The גמרא says that שמואל says that if a lady makes a נדר to not eat two pieces of bread and one of them is considered an עינוי נפש for her and the other is not, he can be מפיר the whole נדר. The ר"ן explains that the reason this works is because if we didn’t allow him to be מפיר the whole נדר then he wouldn’t be able to be מפיר any of it because there is a requirement to always be מפיר the entire נדר and not just part of it. If so, the half that isn’t an עינוי ends up becoming an עינוי because it is standing in the way of allowing her to eat the bread that is a real עינוי. The רא"ש says a different פּשט. He says that since the נדר to not eat two loaves of bread is all one נדר and even the second loaf has a little bit of עינוי נפש, he can be מפיר both.
Both ראשונים ask from the משנה later on דף צ' ע"ב that says if a woman makes a נדר that she is נטלוה מן היהודים the husband can be מפיר only in regards to himself but not in regards to anyone else. We should see from there that שמואל must be incorrect and you can’t be מפיר other parts of a נדר that you don’t have power over. לכאורה the question shouldn’t be a strong question according to the רא"ש since he said that you need the second half to be “קצת עינוי נפש” and in the case נטולהאני מן היהודים it is not "קצת דברים שבינו לבינה" for her to be אסור in תשמיש to other men. (I didn’t see anyone ask this).
The שיטה מקובצת says that the reason the husband can be מפיר both parts of the נדר is because of the principle of נדר שהותר מקצתו הותר כולו. The מנחת שלמה asks that he would think we would say the opposite—instead of saying since I can be מפיר half I can be מפיר the whole thing (or it works ממילא like the שיטה), why don’t we say since I can't be מפיר one half of the נדר I can't be מפיר the other half either? He leaves it as a question. One would think this would not be a question on the ר"ן as the ר"ן explains that they both become עינוי נפש if one impedes access to the other.
The ריטב"א has a fourth approach. He understands that the עינוי נפש food refers to foods she likes, and the non- מתענה loaf refers to a food she doesn’t like. For example, some people don’t like fish and would never eat it even if they didn’t make a נדר. In that case, מתוך that he can be מפיר the נדר on the loaf she likes, he can be מפיר on the loaf she hates. The reason is because there are other people who like it. Moreover, the husband can choose to only(!) be מפיר the one she wouldn’t eat anyway, and even if the נדר on each loaf was said at different times. This would seem to be a big חידוש.
Most ראשונים argue with the explanation of the שיטה מקובצת and say the היתר here is not because of נדר שהותר מקצתו הותר כולו. The question is why not. רבינו אברהם מן ההר explains that we don’t say נדר שהותר מקצתו where you don’t have the power to be מתיר the second half. The רא"ש has a different פּשט. He says that the reason we say נדר שהותר מקצתו הותר כולו is because התרה works by being עוקר הנדר מעיקרו. If so, we assume the person intended on the full נדר and not half a נדר. However, by הפרת נדרים, it is only נעקר מכאן ולהבא so that סברא would not apply.