Near Optimal Bounds for Replacement Paths and Related Problems in the CONGEST Model

TL;DR

This paper establishes near-optimal bounds for fundamental graph problems like Replacement Paths, Minimum Weight Cycle, and All Nodes Shortest Cycles in the CONGEST model, providing both lower and upper bounds and approximation algorithms.

Contribution

It provides near tight bounds for these problems in various graph settings and introduces improved approximation algorithms, advancing understanding of their complexity in the CONGEST model.

Findings

Near linear bounds for directed weighted RPaths, MWC, and ANSC

Near √n bounds for undirected weighted RPaths

A (2 - 1/g)-approximation algorithm for unweighted MWC

Abstract

We present several results in the CONGEST model on round complexity for Replacement Paths (RPaths), Minimum Weight Cycle (MWC), and All Nodes Shortest Cycles (ANSC). We study these fundamental problems in both directed and undirected graphs, both weighted and unweighted. Many of our results are optimal to within a polylog factor: For an $n$-node graph $G$ we establish near linear lower and upper bounds for computing RPaths if $G$ is directed and weighted, and for computing MWC and ANSC if $G$ is weighted, directed or undirected; near $\sqrt{n}$ lower and upper bounds for undirected weighted RPaths; and $\Theta(D)$ bound for undirected unweighted RPaths. We also present lower and upper bounds for approximation versions of these problems, notably a $(2-(1/g))$-approximation algorithm for undirected unweighted MWC that runs in $\tilde{O}(\sqrt{n}+D)$ rounds, improving on the previous best bound of $\tilde{O}(\sqrt{ng}+D)$ rounds, where $g$ is the MWC length. We present a $(1+\epsilon)$-approximation algorithm for directed weighted RPaths, which beats the linear lower bound for exact RPaths.