Geometric Methods in Computer Science

Instructor: Yury Makarychev Course: TTIC 31100 and CMSC 39010-1 Requirements: There will be 3 or 4 homework assignments. There will be no exams. Description: The course covers fundamental concepts in high-dimensional and metric geometry and their applications in computer science. Topics covered include convexity, metric embeddings, dimensionality reduction techniques, Bourgain's theorem, Kirszbraun's theorem, Banach-Mazur distance, Grothendieck's inequality, stochastic decompositions of metric spaces, approximation algorithms for various graph partitioning, coloring, constraint satisfaction problems.

W. Kandinsky: Mild Tension, 1923

Lecture Notes

• Convexity, Convex Hulls, Separating Hyperplanes, Polar Sets
• Brunn–Minkowski, Prékopa–Leindler Inequalities, Applications, Dimension Reduction
• Metric and Normed Spaces
• Bourgain's Theorem
• Sparsest Cut Problem
• Partitioning Metric Spaces
• Dimension Reduction using Gaussian projection

Tentative Syllabus

• Topic 1: Convexity
  convex sets, convex hulls, different definitions and basic properties, extreme points, Carathéodory's theorem, Radon's theorem, Krein–Milman theorem, Helly's theorem, separating hyperplanes, polar sets
• Topic 2: Brunn–Minkowski Inequality, Measure Concentration, Dimension Reduction,
  Brunn–Minkowski inequality, Prékopa–Leindler inequality, Brunn's concavity principle, measure concentration on the sphere, dimension reduction
• Topic 3: Basic Properties of Metric and Normed Spaces, Low-Distortion Metric Embeddings
  metric and normed spaces, Lipschitz maps, distortion, embeddings into L_p and ℓ_p
• Topic 4: Kirszbraun's Theorem, Banach–Mazur Distance Between Normed Spaces
  Kirszbraun's theorem, Banach–Mazur distance between normed spaces, John's theorem and John's ellipsoid, the type and cotype of a Banach space (basic properties)
• Topic 5: Bourgain's Theorem
  Bourgain's theorem
• Topic 6: Approximation Algorithm for Sparsest Cut
  approximation algorithm for Sparsest Cut
• Topic 7: Minimum Balanced Cut, Minimum Linear Arrangement, Sparsest Cut with Non-Uniform Demands, Expanders
  polylogarithmic approximation algorithms for Balanced Cut and Minimum Linear Arrangement, expander graphs, integrality gap for Sparsest Cut and Sparsest Cut with non-uniform demands
• Topic 8: Approximation Algorithms for Minimum Multiway Cut and Minimum Multicut
  approximation algorithms for Minimum Multiway Cut and Minimum Multicut
• Topic 9: Padded Decomposition, Tree Metrics, and Hierarchically Separated Trees (HST)
  padded decomposition, embedding into distributions of dominating trees, HST, applications
• Topic 10: Game Theory: von Neumann's Minimax Theorem and Multiplicative Weight Update Method
  two-player zero-sum games, von Neumann's minimax theorem, multiplicative weight update method
• Topic 11: Räcke's Framework
  Räcke's framework, approximation algorithm for Minimum Bisection
• Topic 12: Semidefinite Programming (SDP)
  positive semidefinite matrices, semidefinite programming, Goemans–Williamson algorithm for Max Cut, Lovasz theta function, and the Karger–Motwani–Sudan algorithm for coloring 3-colorable graphs
• Topic 13: Grothendieck's Inequality and Integer Quadratic Optimization
  Grothendieck's Inequality, Krivine's upper bound on the Grothendieck constant, cut norm, applications in computer science
• Topic 14: Gaussian Correlation Inequality and Discrepancy Minimization
  Gaussian Correlation Inequality, Šidák's theorem, discrepancy minimization, application in SDP approximation algorithms
• Topic 15 (if time permits): Nearest Neighbor Search and Locality-Sensitive Hashing
  approximate nearest neighbor search, locality-sensitive hashing, p-stable random variables