Optimality of Random Regular Graphs in Sparse Network Designs
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 dispersed edge placement can achieve optimal performance under uncertainty.