Diffusion-Robust Optimization over Graphs

2026-05-29

AI Summary

This study addresses robust optimization under edge-weight perturbations in directed graphs, where disturbances propagate along the topological structure subject to node-wise conservation constraints. The authors propose a topology-aware diffusion-based uncertainty model that explicitly incorporates perturbation propagation mechanisms into the network optimization framework. For the first time, this model characterizes how propagation depth, the geometry of uncertainty budgets, and solution structures jointly influence the tractability of robust combinatorial optimization. By integrating tools from graph theory, robust optimization, convex analysis, and computational complexity, the work systematically investigates minimum-cost flow, maximum flow, shortest path, and traveling salesman problems. Key findings include: all convex network flow problems remain polynomially solvable under arbitrary diffusion mechanisms; the shortest path problem exhibits a sharp tractability phase transition; and the robust traveling salesman problem incurs no additional computational complexity beyond its nominal counterpart.

Abstract

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.

Problem

• robust optimization
• graph uncertainty
• diffusion propagation
• combinatorial optimization
• network flows

Innovation

• diffusion-based uncertainty
• robust optimization
• graph networks
• computational tractability
• propagation depth