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

TL;DR

The paper addresses the round complexity of Replacement Paths, Minimum Weight Cycle, and All Nodes Shortest Cycle in the CONGEST model across directed/undirected and weighted/unweighted graphs. It introduces near-optimal lower and upper bounds, including a near-linear bound for directed weighted RPaths via APSP reductions, near-linear lower bounds for directed MWC/ANSC, and sublinear algorithms for undirected RPaths and certain MWC approximations, aided by sampling, skeleton graphs, and routing-table constructions. A central technique is reductions from Set Disjointness to obtain tight lower bounds, complemented by APSP-based reductions to achieve efficient upper bounds; the work also provides exact and approximate constructions of replacement paths and cycles. The results reveal separations between problem variants (e.g., directed weighted versus others) and offer practical implications for fault-tolerant path computations in distributed networks, while outlining several open questions for tightening remaining gaps.

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 $Θ(D)$ 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 $(1+ε)$-approximation algorithm for directed weighted RPaths, which beats the linear lower bound for exact RPaths.

Figures (5)

• Figure 1: Directed weighted RPaths, 2-SiSP lower bound construction
• Figure 2: Directed unweighted RPaths, 2-SiSP lower bound graph $G'$
• Figure 3: Directed weighted RPaths reduction to APSP
• Figure 4: Directed unweighted MWC lower bound construction
• Figure 5: Undirected weighted MWC lower bound construction

Theorems & Definitions (36)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• proof
• proof : Proof of Theorem \ref{['thm:undirrp']}.\ref{['thm:undirrp:lb']} and Theorem \ref{['thm:dirrp']}.\ref{['thm:dirrp:approxlb']}