Relaxation

What is a relaxation?

To "relax" a problem is to remove some of the constraints of the problem, hopefully making the problem easier. The set of feasible solutions to the relaxed problem includes all feasible solutions to the original problem (and the cost functions of the two problems agree on these feasible solutions).

For example, relaxing an IntegerLinearProgram by removing the integrality constraints yields a linear program.

Example: min-cost fractional vertex cover (a relaxation of min-cost vertex cover)

Minimize ∑_v c(v) x(v) subject to:

x(v) ≥ 0 for each vertex v ∈ V.

x(u)+x(v) ≥ 1 for each edge (u,v) ∈ E.

What is the point?

The cost of the optimal solution to the relaxed solution is a lower bound on the cost of the optimal solution to the original problem, because the optimal solution to the original problem is a feasible solution to the relaxed problem.