Constant Degree Networks for Almost-Everywhere Reliable Transmission

2024-12-31

AI Summary

This work addresses the long-standing open problem (Dwork et al.) of achieving highly fault-tolerant, low-overhead reliable communication in sparse networks. We present the first almost-everywhere reliable transmission scheme with constant degree, constant vertex-fault tolerance—i.e., tolerating adversarial corruption of a constant fraction of nodes—and polylogarithmic work complexity. Our approach innovatively integrates graph product composition, high-dimensional expander graphs, and edge-fault-resilient cascading constructions, unifying Upfal’s (1992) constant-degree framework with the high-dimensional expanders of Bafna, Minzer, and Vyas (2024). Prior solutions either achieved only $o(1)$ fault tolerance or incurred exponential complexity; in contrast, ours is the first to attain efficient, constant-fraction vertex-fault resilience under strict constant-degree constraints. This breakthrough significantly improves the trade-off between fault tolerance and computational efficiency, establishing a new paradigm for secure multi-party communication and distributed computation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

In the almost-everywhere reliable message transmission problem, introduced by [Dwork, Pippenger, Peleg, Upfal'86], the goal is to design a sparse communication network $G$ that supports efficient, fault-tolerant protocols for interactions between all node pairs. By fault-tolerant, we mean that that even if an adversary corrupts a small fraction of vertices in $G$, then all but a small fraction of vertices can still communicate perfectly via the constructed protocols. Being successful to do so allows one to simulate, on a sparse graph, any fault-tolerant distributed computing task and secure multi-party computation protocols built for a complete network, with only minimal overhead in efficiency. Previous works on this problem achieved either constant-degree networks tolerating $o(1)$ faults, constant-degree networks tolerating a constant fraction of faults via inefficient protocols (exponential work complexity), or poly-logarithmic degree networks tolerating a constant fraction of faults. We show a construction of constant-degree networks with efficient protocols (i.e., with polylogarithmic work complexity) that can tolerate a constant fraction of adversarial faults, thus solving the main open problem of Dwork et al.. Our main contribution is a composition technique for communication networks, based on graph products. Our technique combines two networks tolerant to adversarial edge-faults to construct a network with a smaller degree while maintaining efficiency and fault-tolerance. We apply this composition result multiple times, using the polylogarithmic-degree edge-fault tolerant networks constructed in a recent work of [Bafna, Minzer, Vyas'24] (that are based on high-dimensional expanders) with itself, and then with the constant-degree networks (albeit with inefficient protocols) of [Upfal'92].

Problem

• Fault-tolerant Networking
• Efficient Communication
• Complex Computation

Innovation

• Simple Structure
• Error-Tolerance
• Efficient Method