Hsin-Hao Su — Page 2

arXiv2014

A Distributed Minimum Cut Approximation Scheme

In this paper, we study the problem of approximating the minimum cut in a distributed message-passing model, the CONGEST model. The minimum cut problem has been well-studied in the context of centralized algorithms. However, there were no known non-trivial algorithms in the distributed model until the recent work of Ghaffari and Kuhn. They gave algorithms for finding cuts of size $O(ε^{-1}λ)$ and $(2+ε)λ$ in $O(D)+\tilde{O}(n^{1/2+ε})$ rounds and $\tilde{O}(D+\sqrt{n})$ rounds respectively, where $λ$ is the size of the minimum cut. This matches the lower bound they provided up to a polylogarithmic factor. Yet, no scheme that achieves $(1+ε)$-approximation ratio is known. We give a distributed algorithm that finds a cut of size $(1+ε)λ$ in $\tilde{O}(D+\sqrt{n})$ time, which is optimal up to polylogarithmic factors.

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arXiv2011

Scaling algorithms for approximate and exact maximum weight matching

The {\em maximum cardinality} and {\em maximum weight matching} problems can be solved in time $\tilde{O}(m\sqrt{n})$, a bound that has resisted improvement despite decades of research. (Here $m$ and $n$ are the number of edges and vertices.) In this article we demonstrate that this "$m\sqrt{n}$ barrier" is extremely fragile, in the following sense. For any $ε>0$, we give an algorithm that computes a $(1-ε)$-approximate maximum weight matching in $O(mε^{-1}\logε^{-1})$ time, that is, optimal {\em linear time} for any fixed $ε$. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error. Our second contribution is a new {\em exact} maximum weight matching algorithm for integer-weighted bipartite graphs that runs in time $O(m\sqrt{n}\log N)$. This improves on the $O(Nm\sqrt{n})$-time and $O(m\sqrt{n}\log(nN))$-time algorithms known since the mid 1980s, for $1\ll \log N \ll \log n$. Here $N$ is the maximum integer edge weight.

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