What do we know about the man and the girl's past life, and what has happened to the quality of their relationship?

We know that the young
woman, called Jig, and the American man now have to "try and have a fine time." It
isn't something that comes easily to them now, though it might once have. Jig tries to say
"bright" things in her effort to lighten the mood of the conversation. The man tells
her that once she has this supposedly "awfully simple operation," they'll "be
fine afterwards. Just like [they] were before [...]. It's the only thing that's made us
unhappy." Therefore, we can see that the couple feels that they used to be happy, but now
they are not, and they struggle to have a good time together.

Further, Jig
asks the man "And if I do it you'll be happy and things will be like they were and you'll
love me?" They both seem to want the relationship to feel the way it
felt before, before the couple got its own white elephant, it seems. Jig points out that the
hills look like white elephants, which is a pretty significant clue (along with references to an
unnamed operation that will restore their...