1 / 24

Discrepancy Minimization by Walking on the Edges

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.

penha
Télécharger la présentation

Discrepancy Minimization by Walking on the Edges

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Discrepancy Minimization by Walking on the Edges Raghu Meka (IAS/DIMACS) Shachar Lovett (IAS)

  2. Discrepancy • Subsets • Color with or -to minimize imbalance 1 2 3 4 5 123 45 3 1 1 0 1

  3. Discrepancy Examples • Fundamental combinatorial concept • Arithmetic Progressions Roth 64: Matousek, Spencer 96:

  4. Discrepancy Examples • Fundamental combinatorial concept • Halfspaces Alexander 90: Matousek 95:

  5. Discrepancy Examples • Fundamental combinatorial concept • Axis-aligned boxes Beck 81: Srinivasan 97:

  6. Why Discrepancy? Complexity theory Communication Complexity Computational Geometry Pseudorandomness Many more!

  7. 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.

  8. 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

  9. 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

  10. Outline Partial coloring Method EDGE-WALK: Geometric picture

  11. Partial Coloring Method Lemma: Can do this in randomized time. Input: Output: • Focus on m = n case.

  12. Outline Partial coloring Method EDGE-WALK: Geometric picture

  13. Discrepancy: Geometric View • Subsets • Color with or -to minimize imbalance 123 45 3 1 1 0 1

  14. Discrepancy: Geometric View • Vectors • Want 123 45

  15. Discrepancy: Geometric View • Vectors • Want Polytope view used earlier by Gluskin’ 88. Goal: Find non-zero lattice points in

  16. 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

  17. Edge-Walk: Algorithm Gaussian random walk in subspaces • Subspace V, rate • Gaussian walk in V Standard normal in V: Orthonormal basis change

  18. Edge-Walk Algorithm Discretization issues: hitting faces • Might not hit face • Slack: face hit if close to it.

  19. Edge-Walk: Algorithm • Input: Vectors • Parameters: For Cube faces nearly hit by . Disc. faces nearly hit by . Subspace orthongal to

  20. Edge-Walk: Intuition Discrepancy faces much farther than cube’s Hit cube more often! 100 1

  21. Summary Spencer’s Theorem Edge-Walk: Algorithmic partial coloring lemma Recurseon unfixed variables

  22. 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?

  23. Thank you

  24. Main Partial Coloring Lemma Th: Given thresholds Can find with 1. 2. Algorithmic partial coloring lemma

More Related