面向几乎处处可靠传输的常数度网络
Constant Degree Networks for Almost-Everywhere Reliable Transmission
AI总结：
本文针对几乎处处可靠传输问题，提出基于图积的组合技术，首次构造出容忍常数比例敌手故障且协议工作复杂度为多对数的常数度网络，解决了 Dwork 等人留下的核心开放问题。
摘要：
在几乎处处可靠消息传输问题中（由 Dwork、Pippenger、Peleg、Upfal 于 1986 年提出），目标是设计一个稀疏通信网络 G，以支持所有节点对之间交互的高效容错协议。所谓容错，是指即使敌手破坏 G 中一小部分顶点，除一小部分顶点外，其余所有顶点仍能通过所构造的协议完美通信。若能做到这一点，就可以在稀疏图上模拟为完全网络构建的任意容错分布式计算任务和安全多方计算协议，且效率开销极小。此前关于该问题的工作要么实现了容忍 o(1) 故障的常数度网络，要么通过低效协议（指数级工作复杂度）实现了容忍常数比例故障的常数度网络，要么实现了容忍常数比例故障的多对数度网络。我们展示了一种具有高效协议（即多对数工作复杂度）的常数度网络构造，该网络能够容忍常数比例的敌手故障，从而解决了 Dwork 等人提出的主要开放问题。我们的主要贡献是一种基于图积的通信网络组合技术。该技术将两个容忍敌手边故障的网络组合起来，构造出度数更小的网络，同时保持效率和容错性。我们多次应用这一组合结果：首先将 Bafna、Minzer、Vyas 于 2024 年最近工作中构造的基于高维扩展器的多对数度边故障容忍网络与其自身组合，然后再与 Upfal 于 1992 年提出的常数度网络（尽管协议低效）组合。
英文摘要：
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].
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