Spectral Techniques for Partitioning Planted Random Graphs

This my final project paper for the class 18.S996: Algorithmic Aspects of Machine Learning, taught in Fall 2013 by Prof. Ankur Moitra. In this note, I will explain the spectral method for partitioning a graph as used in the following two papers: [1] N. Alon, M. Krivelevich, and B. Sudakov, Finding a large hidden clique in a random graph. [4] F. McSherry, Spectral partitioning of random graphs. In [1], the problem is to recover a planted clique of size c √ n in a random graph G(n, 1/2). That is, we first generate a random graph G(n, 1/2), and then randomly put in a clique of size c √ n, and the algorithmic goal is to recover this clique. In [4], variations and extensions of this planted clique problem are considered, including planted bisection and k-coloring. In each case, the graph is generated by first partitioning the vertices into parts of prescribed size (the algorithm doesn’t know which vertices belong to which partition) and then edges are places between parts and within parts using certain prescribed probabilities. The goal is to recover the original partition given the resulting random graph. In [4], McSherry considers probabilities that depend on n, the size of graph, whereas in [1], only the constant 1/2 probability is considered. These problems are all variations of classical NP-hard optimization problems which are even hard to approximate. However, the planted versions turn out to be more amenable, at least in certain parameter regimes. Note that our goal is to recover the planted object/partition, and not necessarily to solve the optimization problem (e.g., finding the largest clique), although these two goals can coincide (as they do in the case of planted clique, since the largest clique in G(n, 1/2) has size (2 + o(1)) log2 n). I will give only a very rough sketch of the proof ideas, without giving much details or calculations.