Suppose someone has parents who are divorced. That someone is close with one parent, but has absolutely no relationship with the other. Must he sit shiva for the estranged parent? Is there any discussion in the poskim about this? I'm familiar with the idea that a couple who did not live harmoniously together do not have to sit shiva for one another, so perhaps there is a similar idea here?
Friedman and Yehuda's The Art of Jewish Pastoral Counseling , p. 96, addresses something to this effect. (While the book aims for an ecumenical pastoral voice, the footnotes always mention Orthodox halacha.) The halachos of shiva still apply, but it doesn't need to be observed the common way. They suggest that the mourner mentally frame it as "grieving for the relationship they wish they could have had", and note that there's absolutely no need for open hours for visitors to come talk about what a wonderful person the departed was (or for the clueless to ask painful questions). The mourner could do the whole thing privately -- but the fact that the relationship was strained doesn't allow them to do things that are normally prohibited.
This is consistent with what I saw when a friend of mine lost a parent with whom the relationship wasn't close. No official visiting hours were publicly given; a few close friends, who could handle the messy reality, were invited.
Mind you, Shulchan Aruch does at least theoretically talk about that a person could be so horrible -- "totally veering off the path of the community" -- that their death would not warrant mourning altogether, but that would require a very serious halachic authority to determine on a case-by-case basis. (It sounds like a wife-beater may qualify, based on something Rabbi Hershel Schachter has said.)
The pastoral-counseling book is citing "Mourning Abusive Parents", Wolowelsky's article in Hakira volume 9; Wolowelsky actually suggests a quiet but clever comeuppance -- normally Kaddish is only said for eleven months, as only the very wicked deserve twelve months in Gehenom. The child of an abusive parent could say it for twelve.
While mourning might seem to be the most natural reaction to the death of a close relative, at times, a person may wish not to mourn, either because their relation to the deceased was strictly biological, or due to estrangement or conflict. We noted above, however, that an adopted child is required to mourn for his or her biological parent, as avelut, especially for a parent, is not dependent upon one's sense of loss, but rather upon an objective biological relationship.
He goes on to note two possible exceptions for abusive parents and someone in the process of divorcing a spouse (or a victim of spousal abuse). Since the situation is complicated and nuanced, he advises consulting with a rabbinic authority in each practical case.
I also checked R Chaim Binyamin Goldberg's Mourning In Halachah, in section 15 ("For whom does one mourn?"). He does NOT list an estranged parent as an exception to the list of those for whom one mourns.
As such it appears halacha prescribes to sit shiva for an estranged parent.