TY - JOUR

T1 - On extracting maximum stable sets in perfect graphs using Lovász's theta function

AU - Yildirim, E. Alper

AU - Fan-Orzechowski, Xiaofei

PY - 2006/3/1

Y1 - 2006/3/1

N2 - 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.

AB - 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.

KW - Lovász's theta function

KW - Maximum stable sets

KW - Perfect graphs

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=33645157104&partnerID=8YFLogxK

U2 - 10.1007/s10589-005-3060-5

DO - 10.1007/s10589-005-3060-5

M3 - Article

AN - SCOPUS:33645157104

VL - 33

SP - 229

EP - 247

JO - Computational optimization and applications

JF - Computational optimization and applications

SN - 0926-6003

IS - 2-3

ER -