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B é la Bollob ás Memphis

B é la Bollob ás Memphis. Guy Kindler Microsoft. Imre Leader Cambridge. Ryan O’Donnell Microsoft. Eliminating cycles in the discrete torus. Q: How many vertices need be deleted to block non-trivial cycles?. (with “L 1 edge structure”).

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B é la Bollob ás Memphis

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  1. Béla BollobásMemphis Guy KindlerMicrosoft Imre LeaderCambridge Ryan O’DonnellMicrosoft Eliminating cycles in the discrete torus

  2. Q: How many vertices need be deleted to block non-trivial cycles?

  3. (with “L1 edge structure”) Q: How many vertices need be deleted to block non-trivial cycles? Upper bound: d ¢ md−1 Upper bound: d ¢ md−1

  4. (with “L1 edge structure”) Q: How many vertices need be deleted to block non-trivial cycles? A: ?¢ md−1 Lower bound: Upper bound: d ¢ md−1 Lower bound: 1 ¢ md−1

  5. Motivation m Lower: Upper: 2 ¢ m ¢ m Best:

  6. tiling of with period (with discretized boundary)

  7. tiling of with period (with discretized boundary)

  8. In dimension d = 2r… m (Hadamard matrix) 0 # of vertices: Theorem 1: upper bound, for d = 2r.

  9. Motivation • “L1 structure”: • [SSZ04]: Asymptotically tight lower bound.(Yields integrality gap for DIRECTED MIN MULTICUT.) • Our Theorem 2: Exactly tight lower bound. • Edge-deletion version: Our original motivation. Connected to quantitative aspects of Raz’s Parallel Repetition Theorem.

  10. Open questions • Obviously, better upper/lower bounds for various versions? (L1 / L1, vertex deletion / edge deletion) • Continuous, Euclidean version: “What tiling of with period has minimal surface area?” Trivial upper bound: d Easy lower bound: No essential improvement known. Best for d = 2:

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