论文标题

查询全球最低削减的复杂性

Query Complexity of Global Minimum Cut

论文作者

Bishnu, Arijit, Ghosh, Arijit, Mishra, Gopinath, Paraashar, Manaswi

论文摘要

在这项工作中,我们通过设计一个随机算法来近似图中的最小切割大小来解决图形的全局最小切割问题的查询复杂性,在该算法中可以通过{\ sc dem},{\ sc neighborn}和{\ sc acecenciency} ceeries来访问图形。 Given $ε\in (0,1)$, the algorithm with high probability outputs an estimate $\hat{t}$ satisfying the following $(1-ε) t \leq \hat{t} \leq (1+ε) t$, where $m$ is the number of edges in the graph and $t$ is the size of minimum cut in the graph.我们的算法使用的本地查询数量为$ \ min \ left \ {m+n,\ frac {m} {t} {t} \ right \} \ mbox {poly} \ left(\ log n,\ log n,\ frac {1}。 Eden和Rosenbaum表明$ω(m/t)$以近似图中最小切割的大小需要许多本地查询。这两个结果共同解决了使用局部查询估算图表中最小切割大小的问题的查询复杂性。 在Eden和Rosenbaum的下限上,我们表明,对于所有$ t \ in \ Mathbb {n} $,$ω(m)$需要本地查询才能确定图中最小切割的最小值是否为$ t $ t $或$ t-2 $。另外,我们表明,对于任何$ t \ in \ mathbb {n} $中的任何$ t \,$ω(m)$ local询问也需要查找所有最小剪切边缘,即使答应输入图的最小缩短尺寸$ t $,也需要查找所有最小剪切边缘。我们的两个下限结果都是随机的,即使我们可以使{\ sc随机边缘}查询除了本地查询之外。

In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like {\sc Degree}, {\sc Neighbor}, and {\sc Adjacency} queries. Given $ε\in (0,1)$, the algorithm with high probability outputs an estimate $\hat{t}$ satisfying the following $(1-ε) t \leq \hat{t} \leq (1+ε) t$, where $m$ is the number of edges in the graph and $t$ is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is $\min\left\{m+n,\frac{m}{t}\right\}\mbox{poly}\left(\log n,\frac{1}ε\right)$ where $n$ is the number of vertices in the graph. Eden and Rosenbaum showed that $Ω(m/t)$ many local queries are required for approximating the size of minimum cut in graphs. These two results together resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries. Building on the lower bound of Eden and Rosenbaum, we show that, for all $t \in \mathbb{N}$, $Ω(m)$ local queries are required to decide if the size of the minimum cut in the graph is $t$ or $t-2$. Also, we show that, for any $t \in \mathbb{N}$, $Ω(m)$ local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size $t$. Both of our lower bound results are randomized, and hold even if we can make {\sc Random Edge} query apart from local queries.

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