Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications

: Ahlswede and Winter [IEEE Trans. Inf. Th. 2002] introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan [JCSS 1988]). As a consequence, we derandomize an efficient construction by Alon and Roichman [RSA 1994] of an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson [STOC 2004]. We also apply these pessimistic estimators to the problem of solving semidefinite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of Ahslwede and Winter. Partially supported by NSF grant CCR-0324906 Supported by an NDSEG Graduate Fellowship and a NSF Graduate Fellowship

• Type: Journal
• Year: 2008
• Where: TOC
• Authors: Avi Wigderson, David Xiao
• Keywords: Graduate Fellowship, Homomorphism Testing Problem, Pessimistic Estimators, TOC 2008