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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

Discrepancy Minimization by Walking on the Edges

Raghu Meka (IAS/DIMACS)

Shachar Lovett (IAS)


Discrepancy
Discrepancy

  • Subsets

  • Color with or -to minimize imbalance

1 2 3 4 5

123 45

3

1

1

0

1


Discrepancy examples
Discrepancy Examples

  • Fundamental combinatorial concept

  • Arithmetic Progressions

Roth 64:

Matousek, Spencer 96:


Discrepancy examples1
Discrepancy Examples

  • Fundamental combinatorial concept

  • Halfspaces

Alexander 90:

Matousek 95:


Discrepancy examples2
Discrepancy Examples

  • Fundamental combinatorial concept

  • Axis-aligned boxes

Beck 81:

Srinivasan 97:


Why discrepancy
Why Discrepancy?

Complexity theory

Communication Complexity

Computational Geometry

Pseudorandomness

Many more!


Spencer s six sigma theorem
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:

  • Tight: Hadamard.


A conjecture and a disproof
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
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
Outline

Partial coloring Method

EDGE-WALK: Geometric picture


Partial coloring method
Partial Coloring Method

Lemma: Can do this in randomized

time.

Input:

Output:

  • Focus on m = n case.


Outline1
Outline

Partial coloring Method

EDGE-WALK: Geometric picture


Discrepancy geometric view
Discrepancy: Geometric View

  • Subsets

  • Color with or -to minimize imbalance

123 45

3

1

1

0

1


Discrepancy geometric view1
Discrepancy: Geometric View

  • Vectors

  • Want

123 45


Discrepancy geometric view2
Discrepancy: Geometric View

  • Vectors

  • Want

Polytope view used earlier by Gluskin’ 88.

Goal: Find non-zero lattice points in


Edge walk
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
Edge-Walk: Algorithm

Gaussian random walk in subspaces

  • Subspace V, rate

  • Gaussian walk in V

Standard normal in V:

Orthonormal basis change


Edge walk algorithm1
Edge-Walk Algorithm

Discretization issues: hitting faces

  • Might not hit face

  • Slack: face hit if close to it.


Edge walk algorithm2
Edge-Walk: Algorithm

  • Input: Vectors

  • Parameters:

For

Cube faces nearly hit by .

Disc. faces nearly hit by .

Subspace orthongal to


Edge walk intuition
Edge-Walk: Intuition

Discrepancy faces much farther than cube’s

Hit cube more often!

100

1


Summary
Summary

Spencer’s Theorem

Edge-Walk: Algorithmic partial coloring lemma

Recurseon unfixed variables


Open problems
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
Main Partial Coloring Lemma

Th: Given thresholds

Can find with

1.

2.

Algorithmic partial coloring lemma


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