The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit -- where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with low edge correlations can achieve optimal performance under uncertainty.

翻译：稀疏网络设计问题在资源分配和运筹学中频繁出现。例如，在生产系统中，采用稀疏流程柔性设计来有效应对不确定需求：目标是构建供需之间最稀疏的二部图，同时使期望满足的需求量达到与完全柔性系统相当的水平。在中途运输中，能够承受随机节点删除后仍保持大规模匹配的稀疏配送路径子图有助于降低运输成本；在此场景下，目标是在节点删除条件下，设计使最大匹配规模与完全连接图保持可比的最稀疏图。稀疏网络设计已被广泛研究，现有最优结果方面，针对二部和单部网络均实现了阶数级最优设计（Chen 等，2015；Feng 等，2024）。然而，确定能实现严格理论极限——即平均度渐近逼近任意图在给定损失水平下可达到的下界——的设计方案，这一问题仍未解决。本文证明随机正则图在二部和单部两种设定下均能达到这一严格最优条件。数值实验进一步验证了此最优性。我们的研究结果为稀疏柔性网络提供了一项实用准则：结合度正则性与低边相关性的设计能在不确定性下实现最优性能。