This is a highly technical matter of great interest in the creation of good social technology, especially social network optimization, but an understanding of it will be useful even to those with no technical background or interests. A specific example is its use in finding the best matches within a small group of people.
Though an important aspect of social technology, matching individuals for romantic liaisons is but one application. Another would be matching individuals to minimize error covariance for good decision making, such as in the justice system.
Bipartite matching thus has a key role to play in making society work.
The most obvious use of weighted bipartite matching is in helping a number of people find spouses, romantic or sexual partners, as explained on my relationship website. Roughly as important, this type of matching can be used in finding jobs for people. Do not underestimate the importance of best friends, however. Some people could be happy with a lower level of romantic compatibility and a less suitable job, but depend on having a true friend. Another social relationship of importance is that between a mentor and protegé.
Together these social connections change the structure of society from a collection of isolated couples to a well connected network. They are an essential part of creating a good social environment.