Approximation Algorithms and Hardness for n-Pairs Shortest Paths and All-Nodes Shortest Cycles
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We study the approximability of two related problems: n-Pairs Shortest Paths (n-PSP), where the goal is to find a shortest path between O(n) prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate n-PSP has been previously studied, mostly in the context of distance oracles. ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of n-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. Our conditional hardness results are based on well-established and believable fine-grained hypotheses. A highlight of our conditional lower bounds results is that under the (k,3)-Hyperclique Hypothesis for any integer k≥ 4, there is no algorithm for unweighted undirected n-PSP with approximation ratio better than 3-6/k that runs in O(n^k/(k-2)-ϵ) time. This is the first known lower bound with approximation ratio higher than 2 for any distance problem except for the ST-Diameter problem, but unlike in n-PSP, the number of vertex pairs one considers in ST-Diameter is much larger than the running time lower bounds. A highlight of our algorithmic results is that one can solve both n-PSP and ANSC in Õ(m+ n^3/2+ϵ) time with approximation factor 2+ϵ (and additive error that is function of ϵ), for any constant ϵ>0. For n-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for n-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms.
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