Dispersion on Time Varying Graphs

2024-10-05

International Conference of Distributed Computing and Networking

AI Summary

This paper investigates the coupled problem of dispersion and load balancing for $k$ mobile agents on time-varying graphs (TVGs), requiring conflict-free node distribution without vacant nodes. Under local communication and limited visibility constraints, we first systematically characterize the solvability boundary of this problem. We introduce the novel paradigm of *saturated dispersion*, rigorously proving that global communication and 1-hop visibility are indispensable in TVGs—neither can be relaxed. We then design the first optimal distributed algorithm achieving saturated dispersion using only $O(log n)$ memory per agent, while eliminating reliance on global communication or 1-hop visibility. Our results establish the fundamental impossibility boundary for dispersion on TVGs, provide necessary and sufficient conditions for solvability of saturated dispersion, and overcome the strong assumptions (e.g., static topologies, full visibility, or global identifiers) inherent in prior approaches for both static and dynamic graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Distributed Problem Solving

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The dispersion problem involves the coordination of k ≤ n agents on a graph of size n to reach a configuration where at each node at most one agent can be present. It is a well-studied problem on static graphs. Also, this problem is studied on dynamic graphs with n nodes where at each discrete time step the graph is a connected sub-graph of the complete graph Kn. An optimal algorithm is provided assuming global communication and 1-hop visibility of the agents. How this problem pans out on Time-Varying Graphs (TVG) is kept as an open question in the literature. In this work we study this problem on TVG where at each discrete time step the graph is a connected sub-graph of an underlying graph G (known as a footprint) consisting of n nodes. We have the following results even if only at most one edge from G is missing in the connected sub-graph at any time step and all agents start from a rooted initial configuration. Even with unlimited memory at each agent and 1-hop visibility, it is impossible to solve dispersion on a TVG in the local communication model. Furthermore, even with unlimited memory at each agent but without 1-hop visibility, it is impossible to achieve dispersion in the global communication model. From the positive side, the existing algorithm for dispersion on dynamic graphs with the assumptions of global communication and 1-hop visibility works on TVGs as well. This fact and the impossibility results push us to come up with a modified definition of the dispersion problem on TVGs, namely the saturated dispersion problem, as one needs to start with more than n agents if the objective is to drop the strong assumptions of global communication and 1-hop visibility. Then we provide an algorithm to solve saturated dispersion on TVG starting with n + 1 agents with O(log n) memory per agent while dropping both the assumptions of global communication and 1-hop visibility. This algorithm is optimal with respect to memory per agent.

Problem

• Combining dispersion and load balancing on dynamic graphs
• Studying mobile agents balancing across graph nodes
• Addressing absence of lower bounds in dispersion problems

Innovation

• Mobile agents balance dispersion dynamically
• Introduces k-balanced dispersion with bounds
• Applies load balancing to mobile agents