Fully Dynamic Algorithms for Minimum Weight Cycle and Related Problems

TL;DR

This paper introduces new fully dynamic algorithms for the minimum weight cycle problem in directed graphs, achieving near-optimal approximate solutions and exact algorithms with improved worst-case update times.

Contribution

It presents the first specialized fully dynamic algorithms for minimum weight cycles, including a near-linear time approximate algorithm and a randomized exact algorithm with better worst-case bounds.

Findings

Deterministic $(1+)$-approximate algorithm with $ ilde{O}(m ext{log}(W)/)$ amortized update time.

Monte Carlo randomized exact algorithm with $ ilde{O}(mn^{2/3})$ worst-case update time.

Generalization of dynamic APSP to multiple-pairs shortest paths with improved update bounds.

Abstract

We consider the directed minimum weight cycle problem in the fully dynamic setting. To the best of our knowledge, so far no fully dynamic algorithms have been designed specifically for the minimum weight cycle problem in general digraphs. One can achieve $\tilde{O}(n^2)$ amortized update time by simply invoking the fully dynamic APSP algorithm of Demetrescu and Italiano [J. ACM'04]. This bound, however, yields no improvement over the trivial recompute-from-scratch algorithm for sparse graphs. Our first contribution is a very simple deterministic $(1+\epsilon)$-approximate algorithm supporting vertex updates (i.e., changing all edges incident to a specified vertex) in conditionally near-optimal $\tilde{O}(m\log{(W)}/\epsilon)$ amortized time for digraphs with real edge weights in $[1,W]$. Using known techniques, the algorithm can be implemented on planar graphs and also gives some new sublinear fully dynamic algorithms maintaining approximate cuts and flows in planar digraphs. Additionally, we show a Monte Carlo randomized exact fully dynamic minimum weight cycle algorithm with $\tilde{O}(mn^{2/3})$ worst-case update that works for real edge weights. To this end, we generalize the exact fully dynamic APSP data structure of Abraham et al. [SODA'17] to solve the ``multiple-pairs shortest paths problem'', where one is interested in computing distances for some $k$ (instead of all $n^2$) fixed source-target pairs after each update. We show that in such a scenario, $\tilde{O}((m+k)n^{2/3})$ worst-case update time is possible.