SIAM Journal on Computing, Volume 50, Issue 4, Page 1155-1199, January 2021.
Among the most important graph parameters is the diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. [Proceedings of SODA 2014, Portland, OR, 2014, pp. 1041--1052] showed that the diameter can be approximated within a multiplicative factor of 3/2 in $\tilde{O}(m^{3/2})$ time. Furthermore, Roditty and Vassilevska W. [Proceedings of STOC '13, New York, ACM, 2013, pp. 515--524] showed that unless the strong exponential time hypothesis (SETH) fails, no $O(n^{2-{\varepsilon}})$ time algorithm can achieve an approximation factor better than 3/2 in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than 3/2. It was, however, completely plausible that a 3/2-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-{\varepsilon}})$ time algorithm can achieve an approximation factor better than 5/3. Another fundamental set of graph parameters is the eccentricities. The eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$. Chechik et al. [Proceedings of SODA 2014, Portland, OR, 2014, pp. 1041--1052] showed that the eccentricities of all vertices can be approximated within a factor of $5/3$ in $\tilde{O}(m^{3/2})$ time and Abboud, Vassilevska W., and Wang [Proceedings of SODA 2016, Arlington, VA, 2016, pp. 377--391] showed that no $O(n^{2-{\varepsilon}})$ algorithm can achieve better than 5/3 approximation in sparse graphs. We show that the runtime of the 5/3 approximation algorithm is also optimal by proving that under SETH, there is no $O(m^{3/2-{\varepsilon}})$ algorithm that achieves a better than 9/5 approximation. We also show that no near-linear time algorithm can achieve a better than 2 approximation for the eccentricities. This is the first lower bound in fine-grained complexity that addresses near-linear time computation. We show that our lower bound for near-linear time algorithms is essentially tight by giving an algorithm that approximates eccentricities within a $2+\delta$ factor in $\tilde{O}(m/\delta)$ time for any $0<\delta<1$. This beats all eccentricity algorithms in Cairo, Grossi, and Rizzi [Proceedings of SODA 2016, Arlington, VA, 2016, pp. 363--376] and is the first constant factor approximation for eccentricities in directed graphs. To establish the above lower bounds we study the $S$-$T$ diameter problem: Given a graph and two subsets $S$ and $T$ of vertices, output the largest distance between a vertex in $S$ and a vertex in $T$. We give new algorithms and show tight lower bounds that serve as a starting point for all other hardness results. Our lower bounds apply only to sparse graphs. We show that for dense graphs, there are near-linear time algorithms for $S$-$T$ diameter, diameter, and eccentricities, with almost the same approximation guarantees as their $\tilde{O}(m^{3/2})$ counterparts, improving upon the best known algorithms for dense graphs.

中文翻译：

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图直径和偏心率的紧逼近边界

SIAM Journal on Computing，第 50 卷，第 4 期，第 1155-1199 页，2021 年 1 月。
其中最重要的图形参数是直径，即任意两个顶点之间的最大距离。没有已知的非常有效的算法来精确计算直径。因此，许多研究致力于该参数的近似速度。Chechik 等人。[SODA 2014 年会刊，俄勒冈州波特兰，2014 年，第 1041--1052 页] 表明直径可以在 $\tilde{O}(m^{3/2}) 中的 3/2 倍增因子内近似$时间。此外，Roditty 和 Vassilevska W. [Proceedings of STOC '13, New York, ACM, 2013, pp. 515--524] 表明，除非强指数时间假设 (SETH) 失败，否则没有 $O(n^{2- {\varepsilon}})$ 时间算法可以在稀疏图中实现优于 3/2 的近似因子。因此，对于近似因子小于 3/2 的稀疏图，上述算法本质上是最佳的。然而，在线性时间内可以进行 3/2 近似是完全合理的。在这项工作中，我们有条件地排除了这种可能性，表明除非 SETH 失败，否则 $O(m^{3/2-{\varepsilon}})$ 时间算法可以实现优于 5/3 的近似因子。另一组基本图形参数是偏心率。顶点 $v$ 的偏心率是 $v$ 与离 $v$ 最远的顶点之间的距离。Chechik 等人。[SODA 2014，波特兰，俄勒冈州，2014 年，第 1041--1052 页]表明所有顶点的偏心率可以在 $\tilde{O}(m^{3/ 2})$ time 和 Abboud、Vassilevska W. 和 Wang [Proceedings of SODA 2016, Arlington, VA, 2016, pp. 377--391] 表明没有 $O(n^{2-{\varepsilon}}) $ 算法在稀疏图中可以达到优于 5/3 的近似。我们证明 5/3 近似算法的运行时间也是最优的，通过证明在 SETH 下，没有 $O(m^{3/2-{\varepsilon}})$ 算法实现优于 9/5近似。我们还表明，没有近线性时间算法可以实现偏心率优于 2 的近似值。这是解决近线性时间计算的细粒度复杂性的第一个下限。我们通过给出一个算法，在 $\tilde{O}(m/\delta)$ 时间为任何 $0<\增量<1$。这击败了 Cairo、Grossi 和 Rizzi [Proceedings of SODA 2016, Arlington, VA, 2016, pp. 363--376] 中的所有偏心率算法，并且是有向图中偏心率的第一个常数因子近似值。为了建立上述下界，我们研究了 $S$-$T$ 直径问题：给定一个图和两个顶点的子集 $S$ 和 $T$，输出 $S$ 中的顶点与$T$。我们提供了新算法并显示了严格的下限，作为所有其他硬度结果的起点。我们的下限仅适用于稀疏图。我们表明，对于密集图，$S$-$T$ 直径、直径和偏心率有接近线性的时间算法，其近似保证与它们​​的 $\tilde{O}(m^{3/2 })$ 对应，改进了最知名的密集图算法。我们提供了新算法并显示了严格的下限，作为所有其他硬度结果的起点。我们的下限仅适用于稀疏图。我们表明，对于密集图，$S$-$T$ 直径、直径和偏心率有接近线性的时间算法，其近似保证与它们​​的 $\tilde{O}(m^{3/2 })$ 对应，改进了最知名的密集图算法。我们提供了新算法并显示了严格的下限，作为所有其他硬度结果的起点。我们的下限仅适用于稀疏图。我们表明，对于密集图，$S$-$T$ 直径、直径和偏心率有接近线性的时间算法，其近似保证与它们​​的 $\tilde{O}(m^{3/2 })$ 对应，改进了最知名的密集图算法。