Approximating Min-Diameter: Standard and Bichromatic
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The min-diameter of a directed graph G is a measure of the largest distance between nodes. It is equal to the maximum min-distance d_min(u,v) across all pairs u,v ∈ V(G), where d_min(u,v) = min(d(u,v), d(v,u)). Our work provides a O(m^1.426n^0.288)-time 3/2-approximation algorithm for min-diameter in DAGs, and a faster O(m^0.713n)-time almost-3/2-approximation variant. (An almost-α-approximation algorithm determines the min-diameter to within a multiplicative factor of α plus constant additive error.) By a conditional lower bound result of [Abboud et al, SODA 2016], a better than 3/2-approximation can't be achieved in truly subquadratic time under the Strong Exponential Time Hypothesis (SETH), so our result is conditionally tight. We additionally obtain a new conditional lower bound for min-diameter approximation in general directed graphs, showing that under SETH, one cannot achieve an approximation factor below 2 in truly subquadratic time. We also present the first study of approximating bichromatic min-diameter, which is the maximum min-distance between oppositely colored vertices in a 2-colored graph.
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