Arc Consistency:

There have been many advances in how constraint solvers search for solutions (remember this means an assignment of a value to each variable in such a way thatno constraint is broken). We look first at a pre-processing step which can greatly improve efficiency by pruning the search space, namely arc-consistency. Following this, we'll look at two search methods, backtracking and forward checking which keep assigning values to variables until a solution is found. Finally, we'll look at some heuristics for improving the efficiency of the solver, namely how to order the choosing of the variables, and how to order the assigning of the values to variables. 

The pre-processing routine for bianry constraints known as  arc-consistency involves calling a pair (xi, xj) an arc and noting that this is an ordered pair, i.e., it is not the same as (xj, xi). Each arc is associated with a single constraint Cij, which constrains variables xi  and xj. We say that the arc (xi, xj) is  consistent  if, for all values a in Di, there is a value b in Dj  such that the assignment xi=a and xj=b satisfies constraint Cij. Note that (xi, xj) being consistent doesn't necessarily mean that (xj,xi) is also consistent. To use this in a pre-processing way, we take every pair of variables and make it arc-consistent. That is, we take each pair (xi,xj) and remove variables from Di  which make it inconsistent, until it becomes consistent. This effectively removes values from the domain of variables, hence prunes the search space and makes it likely that the solver will succeed (or fail to find a solution) more quickly.