Black Hole Search in Dynamic Graphs

TL;DR

This paper investigates the black hole search problem in dynamic graphs, proposing algorithms for different edge removal scenarios and establishing impossibility results, advancing understanding of agent-based search in evolving networks.

Contribution

It introduces algorithms for black hole search in dynamic graphs with edge removals, providing bounds on agent numbers and proving impossibility results under various conditions.

Findings

An algorithm with 9 agents solves BHS in O(|E|^2) time for one-edge removal case.

An improved algorithm with 6 agents solves BHS from a rooted start.

Impossibility of solving BHS with fewer than 2δ_{BH} agents in certain configurations.

Abstract

A black hole is considered to be a dangerous node present in a graph that disposes of any resources that enter that node. Therefore, it is essential to find such a node in the graph. Let a group of agents be present on a graph $G$. The Black Hole Search (BHS) problem aims for at least one agent to survive and terminate after {finding} the black hole. This problem is already studied for specific dynamic graph classes such as rings, cactuses, and tori {where finding the black hole means at least one agent needs to survive and terminate after knowing at least one edge associated with the black hole. In this work, we investigate the problem of BHS for general graphs.} In the dynamic graph, adversary may remove edges at each round keeping the graph connected. We consider two cases: (a) at any round at most one edge can be removed (b) at any round at most $f$ edges can be removed. For both scenarios, we study the problem when the agents start from a rooted initial configuration. We consider each agent has $O(\log n)$ memory and each node has $O(\log n)$ storage. For case (a), we present an algorithm with $9$ agents that solves the problem of BHS in $O(|E|^2)$ time where $|E|$ is the number of edges and $\delta_v$ is the degree of the node $v$ in $G$. We show it is impossible to solve for $2\delta_{BH}$ many agents starting from an arbitrary configuration where $\delta_{BH}$ is the degree of the black hole in $G$. We also provide another improved algorithm that uses $6$ agents from a rooted initial configuration to solve the problem of BHS. For case (b), we provide an algorithm using $6f$ agents to solve the problem of BHS, albeit taking exponential time. We also provide an impossibility result for $2f+1$ agents starting from a rooted initial configuration. This result holds even if unlimited storage is available on each node and the agents have infinite memory.