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

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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

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Discrepancy Examples Fundamental combinatorial concept Arithmetic Progressions

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Discrepancy Examples Fundamental combinatorial concept Halfspaces Alexander 90: Matousek 95:

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

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

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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”

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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.

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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

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Outline 1.Partial coloring Method 2.EDGE-WALK: Geometric picture

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

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Outline 1.Partial coloring Method 2.EDGE-WALK: Geometric picture

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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

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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

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

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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

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

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

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Edge-Walk: Algorithm

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

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

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Open Problems Q: Other applications? General IP’s, Minkowski’s theorem? Some promise: our PCL “stronger” than Beck’s Q:

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Thank you

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Main Partial Coloring Lemma Algorithmic partial coloring lemma

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