arXiv 1506.017992015-06-05cs.DS

半径和直径的逼近和固定参数次二次算法

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter

Amir Abboud, Virginia Vassilevska Williams, Joshua Wang

备注：: Submitted to FOCS 15 on April 2nd

摘要：

半径和直径是图形的基本参数。它们分别被定义为图中偏心率的最小值和最大值，其中顶点的偏心率是从顶点到另一个节点的最大距离。在有向图中，这些问题有几个版本。例如，我们可以选择定义一个节点的偏心率，定义为进入节点的最大距离、离开节点的最大距离、两个方向的和（即往返）等等。所有版本的直径和半径都可以通过求解所有对最短路径（APSP）来求解，然后进行快速后处理步骤。然而，在$n$节点图上求解APSP需要$Ω(n^2)$时间，即使在稀疏图中也是如此，因为需要输出$n^2$距离。出于已知的和新的负面结果的不可能性计算这些措施完全在一般的图形在真正的次二次时间，在合理的假设下，我们寻找\emph{approximation}和\emph{fixed parameter subquadratic}算法，并为他们不存在的原因。我们的成果包括：- 在合理的假设下，适用于大多数具有\emph{optimal}近似保证的Diameter和Radius版本的真正次二次近似算法（给定真正次时间）。特别是，有一个$2$ -近似算法的有向半径与单向距离，运行在$\tilde{O}(m\sqrt{n})$时间，而$(2-δ)$ -近似算法在$O(n^{2-ε})$时间是不可能的。- 在树宽为$k$的图上，我们可以在$2^{O(k\log{k})}n^{1+o(1)}$时间内解决问题。我们表明，这些算法是接近最优的，因为即使是一个$(3/2-δ)$近似算法，运行在时间$2^{o(k)}n^{2-ε}$会反驳似是而非的假设。

英文摘要：

The radius and diameter are fundamental graph parameters. They are defined as the minimum and maximum of the eccentricities in a graph, respectively, where the eccentricity of a vertex is the largest distance from the vertex to another node. In directed graphs, there are several versions of these problems. For instance, one may choose to define the eccentricity of a node in terms of the largest distance into the node, out of the node, the sum of the two directions (i.e. roundtrip) and so on. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. Solving APSP, however, on $n$-node graphs requires $Ω(n^2)$ time even in sparse graphs, as one needs to output $n^2$ distances. Motivated by known and new negative results on the impossibility of computing these measures exactly in general graphs in truly subquadratic time, under plausible assumptions, we search for \emph{approximation} and \emph{fixed parameter subquadratic} algorithms, and for reasons why they do not exist. Our results include: - Truly subquadratic approximation algorithms for most of the versions of Diameter and Radius with \emph{optimal} approximation guarantees (given truly subquadratic time), under plausible assumptions. In particular, there is a $2$-approximation algorithm for directed Radius with one-way distances that runs in $\tilde{O}(m\sqrt{n})$ time, while a $(2-δ)$-approximation algorithm in $O(n^{2-ε})$ time is unlikely. - On graphs with treewidth $k$, we can solve the problems in $2^{O(k\log{k})}n^{1+o(1)}$ time. We show that these algorithms are near optimal since even a $(3/2-δ)$-approximation algorithm that runs in time $2^{o(k)}n^{2-ε}$ would refute the plausible assumptions.

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英文摘要：

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