Core Concepts
This paper presents a new algorithm that computes an O(√(log n)/ε)-approximation for the Sparsest Cut problem using O((nε logO(1) n) · Tmaxflow) expected runtime, where Tmaxflow is the runtime of a maxflow algorithm. The algorithm is parallelizable and can be implemented on O(nε) processors in expected parallel runtime O((logO(1) n) · Tmaxflow).
Abstract
The paper presents a new algorithm for approximating the Sparsest Cut problem, which is a fundamental problem in computer science with many real-world applications. The key contributions are:
Simplification of Sherman's algorithm by avoiding the need to solve a multicommodity flow problem and instead computing "violating paths". This simplifies the algorithm and allows for parallelization.
A new chaining algorithm that is simpler and may yield smaller constants in the approximation ratio compared to prior work.
A parallelizable implementation of the algorithm that can be run on O(nε) processors in expected parallel runtime O((logO(1) n) · Tmaxflow), which is an exponential improvement over the sequential runtime.
The algorithm builds on the Arora-Kale framework for approximately solving SDP relaxations using the Multiplicative Weights Update method. The key steps are:
Procedure Matching(u) that computes a directed matching on a set of vectors based on a given vector u.
Algorithms for "chaining" the matchings computed by Matching(u) to obtain a large set of violating paths.
A final algorithm that combines the violating paths computed in parallel to obtain the final approximation.
The analysis uses measure concentration properties of the Gaussian distribution and a new proof technique that allows for robustness to deletions, enabling the parallelization.