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Monochromatic Triangles, Intermediate Matrix Products, and Convolutions
arXiv - CS - Computational Complexity Pub Date : 2020-09-30 , DOI: arxiv-2009.14479
Andrea Lincoln, Adam Polak, Virginia Vassilevska Williams

The most studied linear algebraic operation, matrix multiplication, has surprisingly fast $O(n^\omega)$ time algorithms for $\omega<2.373$. On the other hand, the $(\min,+)$ matrix product which is at the heart of many fundamental graph problems such as APSP, has received only minor improvements over its brute-force cubic running time and is widely conjectured to require $n^{3-o(1)}$ time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems. For instance, the Min-Max matrix product, the Minimum Witness matrix product, APSP in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle, can all be solved in $\tilde O(n^{(3+\omega)/2})$ time. A similar phenomenon occurs for convolution problems, where analogous intermediate problems can be solved in $\tilde O(n^{1.5})$ time. Can one improve upon the running times for these intermediate problems, in either the matrix product or the convolution world? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way? This paper makes progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem. We also show that a natural convolution variant of monochromatic triangle is fine-grained equivalent to the famous 3SUM problem. As this variant is solvable in $O(n^{1.5})$ time and 3SUM is in $O(n^2)$ time (and is conjectured to require $n^{2-o(1)}$ time), our result gives the first fine-grained equivalence between natural problems of different running times.

中文翻译:

单色三角形、中间矩阵乘积和卷积

研究最多的线性代数运算,矩阵乘法,对于 $\omega<2.373$ 具有惊人的快速 $O(n^\omega)$ 时间算法。另一方面,作为许多基本图问题(例如 APSP)的核心的 $(\min,+)$ 矩阵乘积在其蛮力三次运行时间上仅获得了很小的改进,并且被广泛推测需要 $ n^{3-o(1)}$ 时间。有大量的矩阵乘积和图问题,它们的复杂性似乎位于这两个问题的中间。例如,Min-Max 矩阵乘积、Minimum Witness 矩阵乘积、有向无权图中的 APSP 以及判断边彩色图是否包含单色三角形,都可以在 $\tilde O(n^{(3+ \omega)/2})$ 时间。卷积问题也会出现类似的现象,其中类似的中间问题可以在 $\tilde O(n^{1.5})$ 时间内解决。在矩阵乘积或卷积世界中,可以改进这些中间问题的运行时间吗?或者,人们能否以一种有意义的方式将这些问题相互联系起来,并与其他关键问题联系起来?本文通过提供一个细粒度的减少网络在这些问题上取得了进展。例如,我们展示了有向无权图中的 APSP 和最小见证产品可以减少到最小-最大产品和单色三角形问题的变体。我们还表明,单色三角形的自然卷积变体是细粒度的,相当于著名的 3SUM 问题。由于此变体可在 $O(n^{1.5})$ 时间内求解,而 3SUM 在 $O(n^2)$ 时间内求解(并且推测需要 $n^{2-o(1)}$ 时间) ,
更新日期:2020-10-01
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