New Approximation Algorithms and Reductions for n-Pairs Shortest Paths and All-Nodes Shortest Cycles

In this paper, we focus on two related problems, the n-Pairs Shortest Paths (n-PSP) problem and the All-Nodes Shortest Cycles (ANSC) problem. In the n-PSP problem, given a graph G with n vertices and m edges, as well as a set P ⊆ V × V consisting of at most n pairs of vertices, our objective is to estimate the distances between each pair (u, v) in P. In the ANSC problem, the objective is to find for each node the shortest cycle that includes that particular node. In both problems, we present new algorithms and reductions that enhance the existing solutions in terms of both time complexity and approximation factor. The n-PSP problem has been a subject of study since the 1990s [5], whereas the ANSC problem was introduced by Yuster in 2011 [29]. Recently, Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS 2022] revisited both the n-PSP and ANSC problems, making advancements in the realm of approximation algorithms and conditional lower bounds results. The problems are closely related. Dalirrooyfard et al. proved that for directed graphs, there is a reduction from the n-PSP problem to the ANSC problem that preserves a multiplicative approximation factor. However, for undirected graphs, they demonstrated a reduction only from the exact ANSC problem to the exact n-PSP problem. In other words, no reductions that preserve the approximation ratio in an undirected graph are known in either direction. We present the first reduction for undirected graphs between these problems that assumes an approximation algorithm: We prove that for unweighted graphs, given an algorithm for k-approximation n-PSP where k is a constant, then for any small constant ϵ(k) > 0, one can solve a k + ϵ approximation to the ANSC in comparable time. From the algorithmic side, for undirected weighted graphs, we present an algorithm that solves both n-PSP and ANSC approximation problems with only slight modifications, providing a 2k + 1 approximation for the n-PSP and a k + 2 approximation for the ANSC in O̴ ⁽m^{1− k1} n ^k2) time¹. Moreover, we contribute improvements to many existing approximation algorithms for these problems. For the n-PSP problem, our main result breaks the multiplicative bound of the Thorup-Zwick oracle [26], and gets a (⌈⁴₃^k ⌉ − 1, ⌈⁴₃^k ⌉ − 1)-approximation algorithm in undirected unweighted graphs that runs in O ⁽kmn ^k1 + k²n^{1+ k2)} time. So far, algorithms for n-PSP that run in O̴(mn^1/k) time provide 2k ± O(1) approximation factor, and this is the first result to improve the multiplicative factor beyond 2 (for graphs that are not super sparse). For the ANSC problem, we have developed a 4-multiplicative randomized approximation algorithm that operates in O(m + n¹⁺³/⁴) time by combining two algorithms. The first algorithm finds an almost 2-multiplicative approximation in O(n√m) time. The second algorithm is a novel construction of a fast 2-multiplicative ANSC spanner — a new type of spanner specifically designed to preserve the lengths of the shortest cycles up to a certain stretch factor. For reference, Dalirrooyfard et al. [15] proved that in O(mn ^k ) time, one can solve a 2k−2 approximation 1 for the n-PSP, and a k + ϵ approximation for the ANSC. They also showed that one can solve both n-PSP and ANSC in O̴(m+n³/2+^ϵ) with an approximation factor of a (2+ϵ, f(ϵ)), where ϵ > 0 and f(ϵ) = O(ϵ^−1/ϵ).