Iterative Rounding and Relaxation

In this survey paper we present an iterative method to analyze linear programming formulations for combinatorial optimization problems. This method is introduced by Jain to give a 2-approximation algorithm for the survivable network design problem. First we will present Jain’s method and the necessary background including the uncrossing technique in Section 2. Then we extend the iterative method by a new relaxation step to tackle degree-bounded network design problems, and obtain approximation algorithms with only additive constant errors on the degrees in Section 3. For the minimum bounded degree spanning tree problem, this gives a very simple approximation algorithm with error at most one on the degrees, proving a conjecture of Goemans in Section 4. This method can also be applied to directed graphs, and some recent results on the minimum bounded degree arborescence problem will be highlighted in Section 5. Finally, we discuss how this method provides new proofs of exact linear programming formulations for classical combinatorial optimization problems, and present some new results for the degree bounded matroid problem and the degree bounded submodular flow problem in Section 6.