23.1 Improving Convex Programming Relaxations 23.2.1 Min-cost Non-bipartite Perfect Matching

In past lectures we’ve learned that we write integer linear program (ILP) formulations to model hard problems exactly. But since these programs are too hard to solve, we relax them into linear programs (LPs) and solve those instead. After carefully rounding the solution of an LP relaxation, we attain an approximation of an optimal solution for the original problem. Sometimes, however, our relaxations yield large integrality gaps, hence poor approximation results that we wish to improve upon. One technique for reducing an integrality gap is to modify the ILP by adding “valid” constraints to it, then relaxing the new ILP into an LP, and solving for the new LP. Another technique is to look for higher-order relaxations that can be solved to arbitrary precision in polynomial time, like semidefinite programs.