Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
Approximating a Finite Metric by a Small Number of Tree Metrics
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Parallel repetition: simplifications and the no-signaling case
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Understanding Parallel Repetition Requires Understanding Foams
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Parallel repetition in projection games and a concentration bound
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Spherical Cubes and Rounding in High Dimensions
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A Counterexample to Strong Parallel Repetition
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
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Foam problems are about how to best partition space into bubbles of minimal surface area. We investigate the case where one unit-volume bubble is required to tile d-dimensional space in a periodic fashion according to the standard, cubical lattice. While a cube requires surface area 2d, we construct such a bubble having surface area very close to that of a sphere; that is, proportional to √d (the minimum possible even without the constraint of being periodic). Our method for constructing this "spherical cube" is inspired by foundational questions in the theory of computation related to the concept of hardness amplification. Our methods give new algorithms for "coordinated discretization" of high-dimensional data points, which have near-optimal noise resistance. We also provide the most efficient known cubical foam in three dimensions.