We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy networks. We analyze how such diffusive uncertainty reshapes the computational landscape of robust graph optimization. For convex network problems, such as minimum-cost flow and maximum flow, the resulting formulations remain convex and admit polynomial-time solution methods across all diffusion regimes considered. For combinatorial problems, the effect is more delicate. We focus on two canonical combinatorial graph problems, shortest path and the traveling salesman problem (TSP), which provide complementary benchmarks: shortest path is polynomial-time solvable in the nominal setting, whereas TSP is already NP-hard. We show that, for shortest path, propagation depth induces a sharp transition between tractable and intractable robust counterparts. For the traveling salesman problem, robustness often adds no computational complexity beyond ordinary TSP, because the structure of Hamiltonian cycles makes the fixed-tour adversarial problem collapse to explicit formulas. Together, these results show that topology-aware uncertainty can fundamentally change robust combinatorial optimization, with tractability governed by the interaction between propagation, budget geometry, and the structure of feasible solutions.

翻译：我们引入了一种基于扩散的不确定性模型，用于有向图上的鲁棒优化，其中边权重的扰动沿相邻边传播，并在节点处满足守恒约束。这种拓扑感知结构自然地出现在由流动和局部相互作用引起不确定性的网络化系统中，包括交通、物流、通信和能源网络。我们分析了这种扩散不确定性如何改变鲁棒图优化的计算 landscape。对于凸网络问题，如最小成本流和最大流，最终的形式化表示在所有考虑的扩散情景下仍保持凸性，并允许在多项式时间内求解。对于组合问题，影响则更为微妙。我们聚焦于两个典型的组合图问题：最短路径和旅行商问题，它们提供了互补的基准：最短路径在名义设定下可在多项式时间内求解，而旅行商问题已经是NP难的。我们证明，对于最短路径，传播深度会导致其鲁棒对应问题在可解与难解之间发生尖锐转换。对于旅行商问题，鲁棒性通常不会在普通旅行商问题之上增加额外的计算复杂度，因为哈密顿回路的结构使得固定路径的对抗问题简化为显式公式。这些结果共同表明，拓扑感知的不确定性能够从根本上改变鲁棒组合优化，其可解性由传播、预算几何以及可行解结构之间的相互作用所决定。