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Discrepancy Minimization by Walking on the EdgesPowerPoint Presentation

Discrepancy Minimization by Walking on the Edges

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Discrepancy Minimization by Walking on the Edges

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Discrepancy Minimization by Walking on the Edges

Raghu Meka (IAS/DIMACS)

Shachar Lovett (IAS)

- Subsets
- Color with or -to minimize imbalance

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- Fundamental combinatorial concept

- Arithmetic Progressions

Roth 64:

Matousek, Spencer 96:

- Fundamental combinatorial concept

- Halfspaces

Alexander 90:

Matousek 95:

- Fundamental combinatorial concept

- Axis-aligned boxes

Beck 81:

Srinivasan 97:

Complexity theory

Communication Complexity

Computational Geometry

Pseudorandomness

Many more!

Spencer 85: System with n sets has discrepancy at most .

“Six standard deviations suffice”

- Central result in discrepancy theory.
- Beats random:
- Tight: Hadamard.

Conjecture (Alon, Spencer): No efficient algorithm can find one.

Bansal 10: Can efficiently get discrepancy .

Spencer 85: System with n sets has discrepancy at most .

- Non-constructive pigeon-hole proof

Main: Can efficiently find a coloring with discrepancy

New elemantary constructive proof of Spencer’s result

- Truly constructive
- Algorithmic partial coloring lemma
- Extends to other settings

EDGE-WALK: New algorithmic tool

Partial coloring Method

EDGE-WALK: Geometric picture

Lemma: Can do this in randomized

time.

Input:

Output:

- Focus on m = n case.

Partial coloring Method

EDGE-WALK: Geometric picture

- Subsets
- Color with or -to minimize imbalance

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- Vectors
- Want

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- Vectors
- Want

Polytope view used earlier by Gluskin’ 88.

Goal: Find non-zero lattice points in

Claim: Will find good partial coloring.

- Start at origin
- Gaussian walk until you hit a face
- Gaussian walk within the face

Goal: Find non-zero lattice point in

Gaussian random walk in subspaces

- Subspace V, rate
- Gaussian walk in V

Standard normal in V:

Orthonormal basis change

Discretization issues: hitting faces

- Might not hit face
- Slack: face hit if close to it.

- Input: Vectors
- Parameters:

For

Cube faces nearly hit by .

Disc. faces nearly hit by .

Subspace orthongal to

Discrepancy faces much farther than cube’s

Hit cube more often!

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Spencer’s Theorem

Edge-Walk: Algorithmic partial coloring lemma

Recurseon unfixed variables

- Some promise: our PCL “stronger” than Beck’s

Q:

Beck-Fiala Conjecture 81: Discrepancy for degree t.

Q: Other applications?

General IP’s, Minkowski’s theorem?

Th: Given thresholds

Can find with

1.

2.

Algorithmic partial coloring lemma