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). Discrepancy. Subsets Color with or - to minimize imbalance. 1 2 3 4 5. 1 2 3 4 5. 3. 1. 1. 0. 1. Discrepancy Examples. Fundamental combinatorial concept.

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### Discrepancy Minimization by Walking on the Edges

Raghu Meka (IAS/DIMACS)

Shachar Lovett (IAS)

Discrepancy
• Subsets
• Color with or -to minimize imbalance

1 2 3 4 5

123 45

3

1

1

0

1

Discrepancy Examples
• Fundamental combinatorial concept
• Arithmetic Progressions

Roth 64:

Matousek, Spencer 96:

Discrepancy Examples
• Fundamental combinatorial concept
• Halfspaces

Alexander 90:

Matousek 95:

Discrepancy Examples
• Fundamental combinatorial concept
• Axis-aligned boxes

Beck 81:

Srinivasan 97:

Why Discrepancy?

Complexity theory

Communication Complexity

Computational Geometry

Pseudorandomness

Many more!

Spencer’s Six Sigma Theorem

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

“Six standard deviations suffice”

• Central result in discrepancy theory.
• Beats random:
A Conjecture and a Disproof

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
This Work

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

Outline

Partial coloring Method

EDGE-WALK: Geometric picture

Partial Coloring Method

Lemma: Can do this in randomized

time.

Input:

Output:

• Focus on m = n case.
Outline

Partial coloring Method

EDGE-WALK: Geometric picture

Discrepancy: Geometric View
• Subsets
• Color with or -to minimize imbalance

123 45

3

1

1

0

1

Discrepancy: Geometric View
• Vectors
• Want

123 45

Discrepancy: Geometric View
• Vectors
• Want

Polytope view used earlier by Gluskin’ 88.

Goal: Find non-zero lattice points in

Edge-Walk

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

Edge-Walk: Algorithm

Gaussian random walk in subspaces

• Subspace V, rate
• Gaussian walk in V

Standard normal in V:

Orthonormal basis change

Edge-Walk Algorithm

Discretization issues: hitting faces

• Might not hit face
• Slack: face hit if close to it.
Edge-Walk: Algorithm
• Input: Vectors
• Parameters:

For

Cube faces nearly hit by .

Disc. faces nearly hit by .

Subspace orthongal to

Edge-Walk: Intuition

Discrepancy faces much farther than cube’s

Hit cube more often!

100

1

Summary

Spencer’s Theorem

Edge-Walk: Algorithmic partial coloring lemma

Recurseon unfixed variables

Open Problems
• 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?

Main Partial Coloring Lemma

Th: Given thresholds

Can find with

1.

2.

Algorithmic partial coloring lemma