There are many applications of the tree embedding theorem in approximation and online algorithms. It allows one to reduce certain class of problems on graphs to problems on trees and it is much easier to design algorithms on trees. The ratio one obtains using this approach might not be optimal but gives one a quick upper bound on the approximation ratio. For some problems this is still the best...

• Assume that the input is random, and find an algorithm that will perform well in the average case. For example, the maximum clique problem, which is NP -hard, can actually be solved efficiently assuming a random input because the maximum clique in a randomly chosen graph is small. This assumption is often used in practice, but the problem is that not everyone will agree on whether the input d...

In the study of approximation algorithms, there are many useful techniques that are demonstrated in the analysis of cut problems. Given a graph G = (V,E) with an assignment of weights to edges w : E → R+ a cut is a set E′ ⊆ E, whose removal from the graph will partition the vertices into two disconnected sets. For two vertices, s, t ∈ V , an s− t cut is the one that creates a partition with s a...