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Bounding Non-Trivial Cycles on Discrete Torus: Edge Deletion Theory

This study by Béla Bollobás, Imre Leader, and Ryan O’Donnell from Cambridge and Microsoft explores eliminating cycles on the discrete torus by deleting vertices. The paper presents upper and lower bounds, with a focus on L1 edge structures. Through a theoretical analysis, the researchers address significant questions related to blocking non-trivial cycles. The results contribute to understanding the quantitative aspects of edge deletion in graph theory. Open questions remain for further exploration.

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Bounding Non-Trivial Cycles on Discrete Torus: Edge Deletion Theory

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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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