A sublinear-time randomized parallel algorithm for the maximum clique problem in perfect graphs
SODA '91 Proceedings of the second annual ACM-SIAM symposium on Discrete algorithms
Approximating the independence number via the j -function
Mathematical Programming: Series A and B
On a Representation of the Matching Polytope Via Semidefinite Liftings
Mathematics of Operations Research
A Spectral Bundle Method for Semidefinite Programming
SIAM Journal on Optimization
Computational Experience with Stable Set Relaxations
SIAM Journal on Optimization
A Polynomial Algorithm for Recognizing Perfect Graphs
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Algorithm 875: DSDP5—software for semidefinite programming
ACM Transactions on Mathematical Software (TOMS)
Perfect Constraints Are Tractable
CP '08 Proceedings of the 14th international conference on Principles and Practice of Constraint Programming
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We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the 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.