The Lovasz $\theta$ function, SVMs and finding large dense subgraphs

The Lovasz $\theta$ function of a graph, is a fundamental tool in combinatorial optimization and approximation algorithms. Computing $\theta$ involves solving a SDP and is extremely expensive even for moderately sized graphs. In this paper we establish that the Lovasz $\theta$ function is equivalent to a kernel learning problem related to one class SVM. This interesting connection opens up many opportunities bridging graph theoretic algorithms and machine learning. We show that there exist graphs, which we call $SVM-\theta$ graphs, on which the Lovasz $\theta$ function can be approximated well by a one-class SVM. This leads to a novel use of SVM techniques to solve algorithmic problems in large graphs e.g. identifying a planted clique of size $\Theta({\sqrt{n}})$ in a random graph $G(n,\frac{1}{2})$. A classic approach for this problem involves computing the $\theta$ function, however it is not scalable due to SDP computation. We show that the random graph with a planted clique is an example of $SVM-\theta$ graph, and as a consequence a SVM based approach easily identifies the clique in large graphs and is competitive with the state-of-the-art. Further, we introduce the notion of a ''common orthogonal labeling'' which extends the notion of a ''orthogonal labelling of a single graph (used in defining the $\theta$ function) to multiple graphs. The problem of finding the optimal common orthogonal labelling is cast as a Multiple Kernel Learning problem and is used to identify a large common dense region in multiple graphs. The proposed algorithm achieves an order of magnitude scalability compared to the state of the art.