Approximation Algorithms for Min-Distance Problems in DAGs
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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 Õ(mn) time. So it is natural to resort to approximation algorithms in Õ(mn^1-ϵ) time for some positive ϵ. 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 Õ(m√(n)) time, and showing that any (2-δ)-approximation requires n^2-o(1) time for any δ>0, under the Hitting Set Conjecture. We close the gap, obtaining a 2-approximation algorithm which runs in Õ(m√(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-δ)-approximation algorithm for sparse DAGs requires n^2-o(1) time under SETH. We close this gap for dense DAGs by obtaining a 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^ω - o(1)) time under the high dimensional Orthogonal Vectors Conjecture, where ω is the matrix multiplication exponent.
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