We study the approximability of two related problems on graphs with n nodes
and m edges: 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. We ask the question of whether approximate n-PSP can be
solved faster than by using distance oracles or All Pair Shortest Paths (APSP).
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.
A highlight of our conditional lower bounds results is that for any integer
k≥1, under the combinatorial 4k-clique hypothesis, there is no
combinatorial algorithm for unweighted undirected n-PSP with approximation
ratio better than 1+1/k that runs in O(m2−2/(k+1)n1/(k+1)−ϵ)
time. This nearly matches an upper bound implied by the result of Agarwal
(2014).
A highlight of our algorithmic results is that one can solve both n-PSP and
ANSC in O~(m+n3/2+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.Comment: Abstract truncated to meet arXiv requirement. To appear in FOCS 202