Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs

We present a general method of designing fast approximation algorithms for cut-based minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs while only losing a poly-logarithmic factor in the approximation guarantee. To illustrate the applicability of our paradigm, we focus our attention on the undirected sparsest cut problem with general demands and the balanced separator problem. By a simple use of our framework, we obtain poly-logarithmic approximation algorithms for these problems that run in time close to linear. The main tool behind our result is an efficient procedure that decomposes general graphs into simpler ones while approximately preserving the cut-flow structure. This decomposition is inspired by the cut-based graph decomposition of R\"acke that was developed in the context of oblivious routing schemes, as well as, by the construction of the ultrasparsifiers due to Spiel man and Teng that was employed to preconditioning symmetric diagonally-dominant matrices.

Authors

• Aleksander Madry

Keywords

• Approximation methods
• Approximation algorithms
• Algorithm design and analysis
• Particle separators
• Partitioning algorithms
• Minimization
• Context
• Undirected
• Fast Algorithm
• Algorithm For Problem
• Graph Problems
• Estimation Algorithm
• Minimization Problem
• Impaired Balance
• Efficient Procedure
• Cut Set
• Running Time
• Sparsity
• Efficient Algorithm
• Type Of Approach
• Maximum Flow
• Flow Problem
• Flow Estimation
• Convex Combination
• Nonnegative Function
• Approximate Ratio
• Graph Partitioning
• Flow Algorithm
• Task Of Finding
• Input Graph
• Decomposition Procedure
• Concurrent Problems
• Maximum Flow Rate
• cut-based problems
• generalized sparsest cut
• balanced separator
• fast approximation algorithms
• graph decomposition