In this lecture we introduce the notion of a tree-cotree decomposition for bounded genus graphs (analogous to interdigitating trees in planar graphs) and use it to obtain a spanner for Steiner tree in bounded genus graphs. Together with the contraction decomposition theorem of Lecture 23 and Klein's PTAS framework, this results in an O(n log n) PTAS for Steiner tree in bounded genus graphs.
Afterwards, we draw a bigger picture of all graph classes we have studied so far and take a peek at graph classes beyond H-minor-free graphs, in particular, classes of bounded expansion and nowhere dense classes of graphs. To understand these classes, we introduce the notion of shallow minors.
Indeed, structural graph theory does not end at H-minor-free graphs!
Graphs in the real world tend to have a lot of special structure. In particular, graphs arising on this planet are often planar (or nearly so), meaning that they can be drawn in the plane or a sphere without any (or with few) edge crossings. This class is about efficient algorithms that exploit the structure of planar graphs and related graph classes, specifically graphs embeddable on a surface of low genus and graphs excluding a minor. We will focus on recent results in this exciting area (which all of the instructors actively do research in).
We will organize an optional problem-solving session, during which we can jointly try to solve open problems. In the past, these sessions have led to important new results and published papers, as well as class projects.
Class projects more generally can take the form of formulations of clean, new open problems; implementations of existing algorithms; or well-written descriptions of one or more papers in the area. Projects can be purely mathematical and/or theoretical computer science (algorithmic/complexity theoretic), or purely practical.
The Massachusetts Institute of Technology (MIT), founded in 1861, is located in Cambridge, Massachusetts, and is one of the foremost U.S. institutions in science and technology. It is comprised of five schools and one college, including the renowned School of Engineering and Sloan School of Management, offering Bachelor's, Master's, and Doctorate degrees. Notable alumni include Ben Bernanke, Chairman of the Federal Reserve, Benjamin Netanyahu, prime minister of Israel, and American astronaut "Buzz" Aldrin.