Iterative Methods In Combinatorial Optimization

We describe a simple iterative method for proving a variety of results in combinatorial optimization. It is inspired by Jain's iterative rounding method (FOCS 1998) for designing approx- imation algorithms for survivable network design problems, and augmented with a relaxation idea in the work of Lau, Naor, Salvatipour and Singh (STOC 2007) on designing an approximation al- gorithm for its degree bounded version. At the heart of the method is a counting argument that redistributes tokens from the columns to the rows of an LP extreme point. This token argument was further refined to fractional assignment and redistribution in work of Bansal, Khandekar and Nagarajan on degree-bounded directed network design (STOC 2008). In this presentation, we introduce the method using the assignment problem, describe its applica- tion to showing the integrality of Edmond's characterization (1971) of the spanning tree polyhedron, and then extend the argument to show a simple proof of the Singh and Lau's approximation algo- rithm (STOC 2007) for its degree constrained version, due to Bansal, Khandekar and Nagarajan. We conclude by showing how Jain's original proof can also be simplified by using a fractional token argument (joint work with Nagarajan and Singh). This presentation is extracted from an upcoming monograph on this topic co-authored with Lau and Singh.