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

2. 1 2 3 4 5 Discrepancy 1*11* *11*1 11111 ***11 1*1*1 1 2 3 4 5 1*11* *11*1 11111 ***11 1*1*1 3 1 1 0 1

3. Discrepancy Examples Fundamental combinatorial concept Arithmetic Progressions

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 Central result in discrepancy theory. Beats random: Tight: Hadamard. Spencer 85: System with n sets has discrepancy at most. “Six standard deviations suffice”

8. Conjecture (Alon, Spencer): No efficient algorithm can find one. Bansal 10: Can efficiently get discrepancy. A Conjecture and a Disproof Non-constructive pigeon-hole proof Spencer 85: System with n sets has discrepancy at most.

9. This Work Truly constructive Algorithmic partial coloring lemma Extends to other settings New elemantary constructive proof of Spencer’s result EDGE-WALK: New algorithmic tool

10. 1 2 3 4 5 Beck-Fiala Setting 1*1** *11*1 *1*1* ***1* 1***1

11. Matches Bansal 10

12. Outline 1.Partial coloring Method 2.EDGE-WALK: Geometric picture 3.Analysis of algorithm

13. Partial Coloring Method 1*11* *11*1 11111 ***11 1*1*1 1*11* *11*1 11111 ***11 1*1*1 1 -1 1 1 -1 1*11* *11*1 11111 ***11 1*1*1 1 -1 0 0 0 1*11* *11*1 11111 ***11 1*1*1 1*11* *11*1 11111 ***11 1*1*1 1 1 0

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

15. Outline 1.Partial coloring Method 2.EDGE-WALK: Geometric picture 3.Analysis of algorithm

16. 1*11* *11*1 11111 ***11 1*1*1 Discrepancy: Geometric View 1 1 1 3 1 1 0 1 3 1 1 0 1 1 2 3 4 5

17. 1*11* *11*1 11111 ***11 1*1*1 Discrepancy: Geometric View 1 1 1 3 1 1 0 1 1 2 3 4 5

18. Discrepancy: Geometric View Goal: Find non-zero lattice point inside Polytope view used earlier by Gluskin’ 88.

19. Claim: Will find good partial coloring. Edge-Walk Start at origin Gaussian walk until you hit a face Gaussian walk within the face Goal: Find non-zero lattice point in

20. Edge-Walk: Algorithm Gaussian random walk in subspaces Standard normal in V: Orthonormal basis change

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

22. Edge-Walk: Algorithm

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

24. Outline 1.Partial coloring Method 2.EDGE-WALK: Geometric picture 3.Analysis of algorithm

25. Edge-Walk: Analysis Lem: For with prob 0.1 and

26. Edge-Walk Analysis

27. Claim 1: Never cross polytope

28. Edge-Walk Analysis 100

29. Edge-Walk Analysis Claim 3: Hit many cube faces -

30. Main Partial Coloring Lemma Algorithmic partial coloring lemma

31. Summary 1.Edge-Walk: Algorithmic partial coloring lemma 2.Recurse on unfixed variables Spencer’s Theorem Beck-Fiala setting similar

32. Open Problems Q: Other applications? General IP’s, Minkowski’s theorem? Some promise: our PCL “stronger” than Beck’s Q: Constructive version of Banszczyk’s bound?

33. Thank you