Béla Bollobás Memphis Guy Kindler Microsoft Imre Leader Cambridge Ryan O’Donnell Microsoft.
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Transcript of Béla Bollobás Memphis Guy Kindler Microsoft Imre Leader Cambridge Ryan O’Donnell Microsoft.
Béla BollobásMemphis
Guy KindlerMicrosoft
Imre LeaderCambridge
Ryan O’DonnellMicrosoft
Q: How many
vertices need be
deleted to block
non-trivial cycles?
(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
(with “L1 edge structure”)
Q: How many
vertices need be
deleted to block
non-trivial cycles?
Upper bound: d ¢
md−1
Lower bound: 1 ¢
md−1
A: ? ¢ md−1
Lower bound:
Motivation
Upper:
Lower: m
2 ¢
m
¢
m
Best:
tiling of with period
(with discretized boundary)
tiling of with period
(with discretized boundary)
0
m
# of vertices:
Theorem 1:
upper bound, for d = 2r.
(Hadamard matrix)
In dimension d = 2r…
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.
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: