What is lp relaxation?

LP relaxation (linear programming relaxation) is a method of solving integer programming problems by relaxing the integer constraints to allow non-integer solutions. This is done by converting the integer programming problem into a linear programming problem. The resulting LP problem can then be solved using standard methods of linear programming.

The LP relaxation method is often used as a preprocessing step in solving integer programming problems. This is because the LP relaxation of an integer programming problem provides a lower bound on the optimal value of the integer programming problem. Any feasible solution to the integer programming problem must also be a feasible solution to the LP relaxation problem. Therefore, if the optimal value of the LP relaxation problem is close to the optimal value of the integer programming problem, it suggests that the integer programming problem has a good chance of having a feasible solution.

The LP relaxation method can also be used as a heuristic method for solving integer programming problems. In this case, the LP relaxation is solved to obtain a non-integer solution, which is then rounded to an integer solution. The quality of the rounded solution depends on the quality of the LP relaxation solution.

Overall, the LP relaxation method is a useful tool for solving integer programming problems. It provides a lower bound on the optimal value of the problem and can also be used as a heuristic method for obtaining an integer solution.