Self-Stabilizing Weakly Byzantine Perpetual Gathering of Mobile Agents

TL;DR

The paper addresses gathering of $k$ mobile agents in the presence of $f$ weakly Byzantine agents in synchronous networks, proving two key impossibilities: gathering cannot be solved without global knowledge of $n$ or $k$, and no self-stabilizing gathering is possible when $k\le 2f$ even with full knowledge. It then introduces the perpetual gathering variant and provides a self-stabilizing algorithm that succeeds using only upper bounds $N,K,F,\Lambda_g$, with a time bound of $O(K\cdot F\cdot \Lambda_g\cdot X(N))$ rounds, where $X(n)$ is the time to visit all nodes. The algorithm employs a confidence graph and trust-based seed selection to mitigate Byzantine influence, and leverages the REN rendezvous primitive to enable pairwise meetings that aggregate into a unified group. This work demonstrates that while exact self-stabilizing gathering is impossible under complete global knowledge constraints, a relaxed perpetual gathering problem remains solvable under bounded knowledge, offering a practical route to robust multi-agent coordination under weak Byzantine faults.

Abstract

We study the \emph{Byzantine} gathering problem involving $k$ mobile agents with unique identifiers (IDs), $f$ of which are Byzantine. These agents start the execution of a common algorithm from (possibly different) nodes in an $n$-node network, potentially starting at different times. Once started, the agents operate in synchronous rounds. We focus on \emph{weakly} Byzantine environments, where Byzantine agents can behave arbitrarily but cannot falsify their IDs. The goal is for all \emph{non-Byzantine} agents to eventually terminate at a single node simultaneously. In this paper, we first prove two impossibility results: (1) for any number of non-Byzantine agents, no algorithm can solve this problem without global knowledge of the network size or the number of agents, and (2) no self-stabilizing algorithm exists if $k\leq 2f$ even with $n$, $k$, $f$, and the length $Λ_g$ of the largest ID among IDs of non-Byzantine agents, where the self-stabilizing algorithm enables agents to gather starting from arbitrary (inconsistent) initial states. Next, based on these results, we introduce a \emph{perpetual gathering} problem and propose a self-stabilizing algorithm for this problem. This problem requires that all non-Byzantine agents always be co-located from a certain time onwards. If the agents know $Λ_g$ and upper bounds $N$, $K$, $F$ on $n$, $k$, $f$, the proposed algorithm works in $O(K\cdot F\cdot Λ_g\cdot X(N))$ rounds, where $X(n)$ is the time required to visit all nodes in a $n$-nodes network. Our results indicate that while no algorithm can solve the original self-stabilizing gathering problem for any $k$ and $f$ even with \emph{exact} global knowledge of the network size and the number of agents, the self-stabilizing perpetual gathering problem can always be solved with just upper bounds on this knowledge.

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof