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: