Approximations by Mona Simpson is about always having to settle for what at first seems like a poor substitute for the real thing. However, these "approximations" are the reality and ultimately, especially in terms of Melinda's relationships, they signify something far more genuine than Melinda's false and inconsistent relationship with her father. The story has parallels with Mona Simpson's own life as she admits that she had an idealized picture of her Syrian-born father which he failed to live up to. Simpson is well-known for using her own life experiences to ground her novels.
In the novel, Melinda is a little girl who has to manage her confusion. A self-absorbed, selfish mother and an indifferent father do not help. Melinda thinks she knows what she wants but her aspirations never amount to anything. Every time she thinks that she is getting close, something changes and "then it passed." The cheap hairbands her father gives her symbolize something for Melinda that has no monetary value, and their cost is immaterial to Melinda, who only cares that they came from her father. However, her mother puts it in perspective when she talks about Melinda's father's minimal contribution to her upbringing.
Everything seems to have a value against which it must be measured. This is why it will never be enough unless perceptions change and Melinda and her mother recognize what real value is, how intangible it is and how based on real life experiences, and not imagined, idealistic images, it is. Only at the end does Melinda recognize how shallow her relationship is with her father. She comes to terms with the fact that it is her relationship with her father that is the compromise--the approximation. Her relationship with Jerry is far more real. She is not settling for less by letting Jerry into her life completely. She is getting the real family she deserves.
Therefore, compromise is a theme which is prominent in the novel and it leads to the main theme or message that appearances can be deceiving and that sometimes the thing you think that you aspire to is the poor substitute for the real thing which is right in front of you.