The Constrainedness of Search

We introduce a parameter that measures the “constrainedness” of an ensemble of combinatorial problems. If problems are over-constrained, they are likely to be insoluble. If problems are under-constrained, they are likely to be soluble. This constrainedness parameter generalizes a number of parameters previously used in different NP-complete problem classes. Phase transitions in different NP classes can thus be directly compared. This parameter can also be used in a heuristic to guide search. The heuristic captures the intuition of making the most constrained choice first, since it is often useful to branch into the least constrained subproblem. Many widely disparate heuristics can be seen as minimizing constrainedness.