Approximation Algorithms for Min-Distance Problems in DAGs

The min-distance between two nodes \(u, v\) is defined as the minimum of the distance from \(v\) to \(u\) or from \(u\) to \(v\), and is a natural distance metric in DAGs. As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in \(\tilde{O}(mn)\) time. So it is natural to resort to approximation algorithms in \(\tilde{O}(mn^{1-\epsilon})\) time for some positive \(\epsilon\). Abboud, Vassilevska W., and Wang [SODA 2016] first studied min-distance problems achieving constant factor approximation algorithms on DAGs, obtaining a \(3\)-approximation algorithm for min-radius on DAGs which works in \(\tilde{O}(m\sqrt{n})\) time, and showing that any \((2-\delta)\)-approximation requires \(n^{2-o(1)}\) time for any \(\delta>0\), under the Hitting Set Conjecture. We close the gap, obtaining a \(2\)-approximation algorithm which runs in \(\tilde{O}(m\sqrt{n})\) time. As the lower bound of Abboud et al only works for sparse DAGs, we further show that our algorithm is conditionally tight for dense DAGs using a reduction from Boolean matrix multiplication. Moreover, Abboud et al obtained a linear time \(2\)-approximation algorithm for min-diameter along with a lower bound stating that any \((3/2-\delta)\)-approximation algorithm for sparse DAGs requires \(n^{2-o(1)}\) time under SETH. We close this gap for dense DAGs by obtaining a near-\(3/2\)-approximation algorithm which works in \(O(n^{2.350})\) time and showing that the approximation factor is unlikely to be improved within \(O(n^{\omega - o(1)})\) time under the high dimensional Orthogonal Vectors Conjecture, where \(\omega\) is the matrix multiplication exponent.