预算约束下的图增强用于基于基尔霍夫指数最小化的鲁棒网络设计
Budget-Constrained Graph Augmentation for Robust Network Design via Kirchhoff Index Minimization
- Norwegian University of Science and Technology（挪威科技大学）
- Aalto University（阿尔托大学）
- SRM University-AP（SRM大学安得拉邦校区）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对预算约束下的网络鲁棒性增强问题，提出基于基尔霍夫指数最小化的混合整数与半定松弛方法，并开发可扩展贪心算法，实验验证了成本机制对链路选择的影响。
AI中文摘要：
增强已部署网络对故障和中断的鲁棒性对于可靠运行至关重要。这需要在有限资源下决定安装哪些新链路以及如何对其加权。我们通过基尔霍夫指数（即总有效电阻）这一全局连通性的谱度量来研究该问题。由此产生的增强问题将离散候选边选择与连续权重分配耦合，并考虑异构的单位部署成本、总预算以及精确基数约束。对于固定的加权基础图，这产生了一个混合整数规划形式和一个半定松弛，其最优值下界于混合整数最优值。我们将该松弛转化为锥规划，并使用齐次自对偶嵌入和一阶算子分裂进行数值求解。通过舍入和修复程序恢复可行的离散设计，并根据数值半定规划（SDP）基准的验后间隙估计进行评估。作为一种可扩展的替代方案，我们开发了一种基于秩一拉普拉斯更新和双调和距离缓存的精确-k、预算可行的贪心启发式算法，并通过贝尔曼值到目标基准和保守的谱下界来解释其进展，该下界用于局部策略比率。在合成和真实基础设施网络上进行的实验，涵盖不同图规模、预算、权重分布和成本机制，结果表明：在固定预算下，距离成比例的成本限制了可实现的电阻降低，并将安装的电导转移到较短链路上，相对于统一单位成本而言。
英文摘要：
Enhancing the robustness of deployed networks against failures and disruptions is critical for reliable operation. This requires deciding which new links to install and how strongly to weight them under limited resources. We study this problem through the Kirchhoff index, or total effective resistance, a spectral measure of global connectivity. The resulting augmentation problem couples discrete candidate-edge selection with continuous weight allocation under heterogeneous per-unit deployment costs, a total budget, and an exact-cardinality constraint. For a fixed weighted base graph, this yields a mixed-integer formulation and a semidefinite relaxation whose optimum lower-bounds the mixed-integer optimum. We cast the relaxation as a cone program and solve it numerically using a homogeneous self-dual embedding and first-order operator splitting. Feasible discrete designs are recovered through rounding-and-repair procedures and assessed by \emph{a posteriori} gap estimates relative to the numerical semidefinite program (SDP) benchmark. As a scalable alternative, we develop an exact-$k$, budget-feasible greedy heuristic built on rank-one Laplacian updates and biharmonic-distance caching, and interpret its progress through a Bellman value-to-go benchmark with a conservative spectral lower bound on the local policy ratio. Experiments on synthetic and real infrastructure networks across graph sizes, budgets, weight distributions, and cost regimes show that, under fixed budgets, distance-proportional costs limit the achievable resistance reduction and shift installed conductance toward shorter links relative to uniform per-unit costs.
