更可信假设下三角形问题的困难性：实数APSP、实数3SUM和OV的约化
Hardness for Triangle Problems under Even More Believable Hypotheses: Reductions from Real APSP, Real 3SUM, and OV
摘要：
$3$ SUM假说、APSP假说和SETH假说是细粒度复杂性的三个主要假说。到目前为止，在该领域内，前两个假设主要是关于Word RAM计算模型中的整数输入。“实APSP”和“实$3$ SUM”假设断言APSP和$3$ SUM假设在实RAM模型的合理版本中对实值输入成立，甚至比它们的整数对应物更可信。在非常可信的假设下，至少有一个$3$ SUM假设，SENAPSP假设或SETH是真的，Abboud，Vassilevska W。和Yu [STOC 2015]表明，一个名为Triangle Collection的问题需要在$n$节点图上花费$n^{3-o(1)}$时间。我们的主要结果是一个非平凡的下限略推广三角形集合，称为全彩色对三角形集合，在更可信的假设下，至少有一个真正的$3$总和，真正的APSP，和OV假设是真的。结合轻微的修改前减少，我们得到多项式条件下界的问题，如（静态）ST-最大流量问题和动态最大流量，现在在新的较弱的假设。我们的主要结果是建立在以下两条减少线。* 实APSP和实$3$ SUM硬度用于全边稀疏三角形问题。以前的减少只从这些问题的整数变量。* 布尔矩阵乘法问题变体的真实APSP和OV难度。一路上，我们证明了三角形集合等价于问题的一个更简单的限制版本，简化了先前的工作。我们的技术也有其他有趣的影响，如超线性下限的所有数字$3$ SUM的基础上，真正的$3$ SUM假设，和一个严格的下限的字符串匹配问题的基础上的OV假设。
英文摘要：
The $3$SUM hypothesis, the APSP hypothesis and SETH are the three main hypotheses in fine-grained complexity. So far, within the area, the first two hypotheses have mainly been about integer inputs in the Word RAM model of computation. The "Real APSP" and "Real $3$SUM" hypotheses, which assert that the APSP and $3$SUM hypotheses hold for real-valued inputs in a reasonable version of the Real RAM model, are even more believable than their integer counterparts. Under the very believable hypothesis that at least one of the Integer $3$SUM hypothesis, Integer APSP hypothesis or SETH is true, Abboud, Vassilevska W. and Yu [STOC 2015] showed that a problem called Triangle Collection requires $n^{3-o(1)}$ time on an $n$-node graph. Our main result is a nontrivial lower bound for a slight generalization of Triangle Collection, called All-Color-Pairs Triangle Collection, under the even more believable hypothesis that at least one of the Real $3$SUM, the Real APSP, and the OV hypotheses is true. Combined with slight modifications of prior reductions, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic Max Flow, now under the new weaker hypothesis. Our main result is built on the following two lines of reductions. * Real APSP and Real $3$SUM hardness for the All-Edges Sparse Triangle problem. Prior reductions only worked from the integer variants of these problems. * Real APSP and OV hardness for a variant of the Boolean Matrix Multiplication problem. Along the way we show that Triangle Collection is equivalent to a simpler restricted version of the problem, simplifying prior work. Our techniques also have other interesting implications, such as a super-linear lower bound of Integer All-Numbers $3$SUM based on the Real $3$SUM hypothesis, and a tight lower bound for a string matching problem based on the OV hypothesis.
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