what kind of algorithm is this (stable marriage variation)?

Question

I have a set of objects (between 1 and roughly 500). Each object is compatible with certain (zero or more) other objects from that same set.

Can anyone give me some pointers as to how to determine the best way to create pairs of objects that are compatible with each other so that most of the objects in the set are paired?

Answer 1

You're looking for a maximum matching in a general graph. As opposed to the stable marriage problem with which you are familiar, in the maximum matching problem the input graph is not necessarily bipartite. There is no notion of stability (as vertices do not rank their compatible options) and what you're looking for is a subset of the edges of the graph such that no two edges share a common vertex (a.k.a., a matching). You're trying to construct that matching which contains the maximum possible number of edges.

Luckily, the problem of finding a maximum matching in a general graph can be solved in polynomial time using Edmond's matching algorithm (also known as the blossom algorithm because of how it contracts blossoms (odd cycles) into single vertices). The time complexity of Edmond's matching algorithm is O(E•V^2). While not very efficient, I believe this is good enough for the relatively small graphs you're dealing with. You don't even have to implement it from scratch by yourself as there's an open source Java implementation of Edmond's algorithm you can use. However, if you're interested in the state of the art, you can use the most efficient algorithm known for the problem which runs in O(E•sqrt(V)).

If the vertex compatibility of your input is not dichotomous (that is, each vertex has a ranking specifying its preferences among its neighbors), you can add corresponding weights to the edges to accommodate for the preference profile and use the variation of Edmond's algorithm for weighted graphs.