Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
SIAM Journal on Computing
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Consequences and Limits of Nonlocal Strategies
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS
Computational Complexity
A note on the /spl theta/ number of Lovasz and the generalized Delsarte bound
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
On the Shannon capacity of a graph
IEEE Transactions on Information Theory
Unique Games with Entangled Provers Are Easy
SIAM Journal on Computing
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In recent years we witness the proliferation of semidefinite programming bounds in combinatorial optimization [1,5,8], quantum computing [9,2,3,6,4] and even in complexity theory [7]. Examples to such bounds include the semidefinite relaxation for the maximal cut problem [5], and the quantum value of multi-prover interactive games [3,4]. The first semidefinite programming bound, which gained fame, arose in the late seventies and was due to László Lovász [11], who used his theta number to compute the Shannon capacity of the five cycle graph. As in Lovász's upper bound proof for the Shannon capacity and in other situations the key observation is often the fact that the new parameter in question is multiplicative with respect to the product of the problem instances. In a recent result R. Cleve, W. Slofstra, F. Unger and S. Upadhyay show that the quantum value of XOR games multiply under parallel composition [4]. This result together with [3] strengthens the parallel repetition theorem of Ran Raz [12] for XOR games. Our goal is to classify those semidefinite programming instances for which the optimum is multiplicative under a naturally defined product operation. The product operation we define generalizes the ones used in [11] and [4]. We find conditions under which the product rule always holds and give examples for cases when the product rule does not hold.