9.5 Approximation Algorithms in Geometry

Approximation algorithms in geometry tackle complex problems by finding near-optimal solutions efficiently. These algorithms provide practical approaches to NP-hard problems like the Traveling Salesman Problem, Steiner Tree, and Facility Location.

Advanced techniques like Polynomial-Time Approximation Schemes and coresets push the boundaries of what's possible. They offer flexible trade-offs between solution quality and running time, making geometric optimization more accessible in real-world applications.

Approximation Algorithms for Optimization Problems

Approximation Ratio and Traveling Salesman Problem

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• Approximation ratio measures algorithm performance by comparing its solution to optimal solution
• Calculated as ratio between approximate solution value and optimal solution value
• Closer to 1 indicates better approximation
• Euclidean traveling salesman problem seeks shortest tour visiting all points in plane
• NP-hard problem, approximation algorithms provide near-optimal solutions
• Christofides algorithm achieves 1.5-approximation ratio for metric TSP
  • Constructs minimum spanning tree
  • Adds minimum-weight perfect matching on odd-degree vertices
  • Forms Eulerian tour and shortcutting yields approximate TSP tour
• 2-approximation algorithm for Euclidean TSP
  • Doubles each edge in minimum spanning tree
  • Creates Eulerian circuit
  • Shortcutting produces tour at most twice optimal length

Steiner Tree and Facility Location Problems

• Steiner tree problem connects given set of points using minimum total edge length
• Differs from minimum spanning tree by allowing additional points (Steiner points)
• NP-hard problem, approximation algorithms provide near-optimal solutions
• MST-based 2-approximation algorithm for Steiner tree
  • Constructs minimum spanning tree on given points
  • Approximates optimal Steiner tree within factor of 2
• Facility location problem optimizes placement of facilities to serve customers
• Seeks balance between facility opening costs and connection costs to customers
• Metric facility location admits constant-factor approximation algorithms
• Primal-dual schema yields 3-approximation algorithm for metric facility location
  • Iteratively raises dual variables
  • Opens facilities and connects clients based on dual solution
  • Provides trade-off between facility and connection costs

Geometric Set Cover

• Geometric set cover problem involves covering points with geometric shapes
• Aims to minimize number of shapes used to cover all points
• NP-hard problem, approximation algorithms provide efficient solutions
• ε-net method for geometric set cover
  • Constructs small subset (ε-net) that intersects all large sets
  • Iteratively selects shapes to cover remaining points
  • Achieves O(log OPT) approximation for many geometric set systems
• Local search algorithms for geometric set cover
  • Iteratively improve solution by local modifications
  • Achieve constant-factor approximations for some geometric set cover variants (disks, rectangles)

Advanced Techniques in Approximation Algorithms

Polynomial-Time Approximation Schemes (PTAS)

• PTAS provides (1 + ε)-approximation for any ε > 0
• Running time polynomial in input size but may be exponential in 1/ε
• Allows trade-off between solution quality and running time
• Shifting strategy for geometric PTAS
  • Partitions space into grids of different sizes
  • Solves subproblems within grid cells
  • Combines solutions to obtain overall approximation
• PTAS for Euclidean TSP in the plane
  • Partitions plane into squares
  • Computes optimal tours within squares
  • Connects square tours to form overall tour
  • Achieves (1 + ε)-approximation for any ε > 0
• Local search PTAS for geometric problems
  • Iteratively improves solution by local modifications
  • Analyzes locality gap to bound approximation ratio
  • Applies to problems like k-median, facility location in low dimensions

Coresets and Their Applications

• Coreset reduces large input to small representative subset
• Preserves approximate solution quality for optimization problem
• Enables faster algorithms by operating on smaller input
• Coreset construction for k-means clustering
  • Samples points proportional to their contributions to clustering cost
  • Assigns weights to sampled points
  • Provides (1 + ε)-approximation with high probability
• Streaming and distributed algorithms using coresets
  • Maintains small coreset in streaming setting
  • Merges coresets from distributed data sources
  • Enables approximation algorithms in big data scenarios
• Geometric problems amenable to coreset techniques
  • Shape fitting (minimum enclosing ball, cylinder)
  • Clustering (k-means, k-median)
  • Extent measures (diameter, width)
• Trade-off between coreset size and approximation quality
  • Smaller coresets allow faster algorithms but may sacrifice accuracy
  • Careful analysis balances size and approximation guarantees