有向图围长的连续最优近似算法
Conditionally optimal approximation algorithms for the girth of a directed graph
摘要：
众所周知，一个比$2$ -近似算法更好的稠密有向无权图的围长需要$n^{3-o(1)}$时间，除非使用快速矩阵乘法。同时，在$O(mn^{1-ε})$时间中运行的组合算法的最佳已知近似因子（由Alfrehik et al.）是$3$。正确答案是$2$还是$3$？本文的主要结果是有向图的一个（条件）紧逼近算法。首先，我们表明，根据一个流行的硬度假设，任何算法，即使是一个利用快速矩阵乘法，将需要至少$mn^{1-o(1)}$时间的一些稀疏$m$，如果它实现了$(2-ε)$ -近似任何$ε>0$。其次，我们给出了运行在$\tilde{O}(mn^{3/4})$时间内的无权图围长的$2$ -近似算法，以及运行在$\tilde{O}(m\sqrt n)$时间内的加权图围长的$(2+ε)$ -近似算法（对于任何$ε>0$）。我们的算法是组合的。我们还得到了周长运行在$\tilde{O}(mn^{\sqrt{2}-1})$时间的$(4+ε)$ -近似，提高了以前最好的$\tilde{O}(m\sqrt n)$运行时间的Escherhik等人。最后，我们考虑了往返spectrum的计算。在$\tilde{O}(m\sqrt n/ε^2)$时间内，我们得到了$\tilde{O}(n^{1.5}/ε^2)$边上的$(5+ε)$ -近似往返图.对于相同的运行时间，这改进了先前的近似因子$(8+ε)$。
英文摘要：
It is known that a better than $2$-approximation algorithm for the girth in dense directed unweighted graphs needs $n^{3-o(1)}$ time unless one uses fast matrix multiplication. Meanwhile, the best known approximation factor for a combinatorial algorithm running in $O(mn^{1-ε})$ time (by Chechik et al.) is $3$. Is the true answer $2$ or $3$? The main result of this paper is a (conditionally) tight approximation algorithm for directed graphs. First, we show that under a popular hardness assumption, any algorithm, even one that exploits fast matrix multiplication, would need to take at least $mn^{1-o(1)}$ time for some sparsity $m$ if it achieves a $(2-ε)$-approximation for any $ε>0$. Second we give a $2$-approximation algorithm for the girth of unweighted graphs running in $\tilde{O}(mn^{3/4})$ time, and a $(2+ε)$-approximation algorithm (for any $ε>0$) that works in weighted graphs and runs in $\tilde{O}(m\sqrt n)$ time. Our algorithms are combinatorial. We also obtain a $(4+ε)$-approximation of the girth running in $\tilde{O}(mn^{\sqrt{2}-1})$ time, improving upon the previous best $\tilde{O}(m\sqrt n)$ running time by Chechik et al. Finally, we consider the computation of roundtrip spanners. We obtain a $(5+ε)$-approximate roundtrip spanner on $\tilde{O}(n^{1.5}/ε^2)$ edges in $\tilde{O}(m\sqrt n/ε^2)$ time. This improves upon the previous approximation factor $(8+ε)$ of Chechik et al. for the same running time.
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