On a positive semidefinite relaxation of the cut polytope

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:36200

Laurent, M; Poljak, S

We study the set L{sub n}, which consists of the symmetric positive semidefinite n {times} n matrices whose diagonal entries are all equal to 1. L{sub n} is a (nonpolyhedral) convex body whose vertices correspond to the cuts of the complete graph on n nodes. Hence, optimization over L{sub n} yields an approximation for the max-cut problem. A recent result of Goemans and Williamson shows that the worst case ratio for this approximation is equal to 1.138. Our main motivation for investigating L{sub n} comes from combinatorial optimization, namely from approximating the max-cut problem. Moreover, due to the positive semidefinite constraints, one can optimize over L{sub n} in polynomial time. Optimizing over L{sub n} provides a good approximation for the max-cut problem. On the other hand, L{sub n} still inherits the difficult structure of the underlying combinatorial problem. In particular, deciding whether the optimum of the problem min (tr(CX) {vert_bar} X {element_of} L{sub n}) is reached in a vertex of L{sub n} is an NP-hard problem. This result follows from the complete characterization of the matrices C of the form C = bb{sup t} for some vector b, for which the optimum of the above program is reached in a vertex. We also show that the worst case ratio for these objective functions is equal to 1.125. We need, in particular, the description of the polar of L{sub n} and of the normal cone to a point of L{sub n}.

• OSTI ID: 36200
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
COMPUTER NETWORKS
OPTIMIZATION
PARALLEL PROCESSING
STOCHASTIC PROCESSES