TY - GEN

T1 - Solution of the propeller conjecture in ℝ 3

AU - Heilman, Steven

AU - Jagannath, Aukosh

AU - Naor, Assaf

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - It is shown that every measurable partition {A 1,..., A k} of ℝ 3 satisfies (Equation Presented) Let {P 1,P 2,P 3} be the partition of ℝ 2 into 120° sectors centered at the origin. The bound (1) is sharp, with equality holding if A i=P i x ℝ for i ∈ {1,2,3} and A i = ∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.

AB - It is shown that every measurable partition {A 1,..., A k} of ℝ 3 satisfies (Equation Presented) Let {P 1,P 2,P 3} be the partition of ℝ 2 into 120° sectors centered at the origin. The bound (1) is sharp, with equality holding if A i=P i x ℝ for i ∈ {1,2,3} and A i = ∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.

KW - grothendieck inequalities

KW - kernel clustering

KW - semidefinite programming

KW - unique games hardness

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U2 - 10.1145/2213977.2214003

DO - 10.1145/2213977.2214003

M3 - Conference contribution

AN - SCOPUS:84862635715

SN - 9781450312455

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 269

EP - 276

BT - STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing

T2 - 44th Annual ACM Symposium on Theory of Computing, STOC '12

Y2 - 19 May 2012 through 22 May 2012

ER -