On Extracting Maximum Stable Sets in Perfect Graphs Using
Lov'asz's Theta Function

by

Emre Alper Yildirim
Bilkent University

We study the maximum stable set problem. For a given graph, we develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using Lov'asz's theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality.

The talk will be mostly self-contained, describing the derivation of Lov'asz's theta function. No background will be assumed.

This is joint work with Xiaofei Fan-Orzechowski, Department of Applied Mathematics and Statistics, Stony Brook University.