Approximation Algorithms Based on the Primal-Dual Method

The primal-dual method (or primal-dual schema) is another means of solving linear programs. The basic idea of this method is to start from a feasible solution y to the dual program, then attempt to find a feasible solution x to the primal program that satisfies the complementary slackness conditions. If such an x cannot be found, it turns out that we can find a better y in terms of its objective value. Then, another iteration is started. The above idea can also be modified to design approximation algorithms. An approximate solution to the primal IP and a feasible solution to the dual LP can be constructed simultaneously and improved step by step. In the end, the approximate solution can be compared with the dual feasible solution to estimate the approximation ratio. One of the key strengths of this method is that it often allows for a combinatorial algorithm (based on the primal/dual view) which is very efficient.