“Within computer science and operations research, many combinatorial optimization problems are computationally intractable to solve exactly (to optimality). Many such problems do admit fast (polynomial time) approximation algorithms – that is, algorithms that are guaranteed to return an approximately optimal solution given any input.
Randomized rounding (Raghavan & Tompson 1987) is a widely used approach for designing and analyzing such approximation algorithms. The basic idea is to use the probabilistic method to convert an optimal solution of a relaxation of the problem into an approximately optimal solution to the original problem.” – from Wikipedia
In the following we prove the correctness of randomized rounding on LP, given the optimal value.