A simpler and parallelizable O(√(log n))-approximation algorithm for Sparsest Cut
arxiv(2023)
摘要
Currently, the best known tradeoff between approximation ratio and complexity
for the Sparsest Cut problem is achieved by the algorithm in [Sherman, FOCS
2009]: it computes O(√((log n)/ε))-approximation using
O(n^εlog^O(1)n) maxflows for any ε∈[Θ(1/log
n),Θ(1)]. It works by solving the SDP relaxation of [Arora-Rao-Vazirani,
STOC 2004] using the Multiplicative Weights Update algorithm (MW) of
[Arora-Kale, JACM 2016]. To implement one MW step, Sherman approximately solves
a multicommodity flow problem using another application of MW. Nested MW steps
are solved via a certain “chaining” algorithm that combines results of
multiple calls to the maxflow algorithm. We present an alternative approach
that avoids solving the multicommodity flow problem and instead computes
“violating paths”. This simplifies Sherman's algorithm by removing a need for
a nested application of MW, and also allows parallelization: we show how to
compute O(√((log n)/ε))-approximation via O(log^O(1)n)
maxflows using O(n^ε) processors. We also revisit Sherman's
chaining algorithm, and present a simpler version together with a new analysis.
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