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 Räcke that was developed in the context of oblivious routing schemes, as well as, by the construction of the ultrasparsifiers due to Spielman and Teng that was employed to preconditioning symmetric diagonally-dominant matrices. The captured by the above-mentioned problems task of partitioning a graph into relatively large pieces while minimizing the number of edges cut is a fundamental combinatorial problem (see [1] ) with a long history of
Informally a hierarchical decomposition of a graph G is a set of trees for which the leaves correspond to the vertices of G, and for which the structure of their cuts approximate the cuts in G. Graph decompositions have been studied in the context of oblivious routing schemes (see REF for a survey).
Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Mathematics Computer Science