We study the problem of network robustness under consensus dynamics. We first show that the consensus interdiction problem (CIP), in which the goal is to maximize the convergence time of consensus dynamics subject to removing limited network edges, can be cast as an effective resistance interdiction problem (ERIP). We then show that ERIP is strongly NP-hard, even for bipartite graphs of diameter three with fixed source/sink edges. We establish the same hardness result for the CIP, hence correcting some claims in the existing literature. We then show that both ERIP and CIP do not admit a polynomial-time approximation scheme, and moreover, they cannot be approximated up to a (nearly) polynomial factor assuming exponential time hypothesis. Finally, using a quadratic program formulation, we devise a polynomial-time $n^4$-approximation algorithm for ERIP that only depends on the number of nodes $n$ and is independent of the size of edge resistances. We also develop an iterative heuristic approximation algorithm to find a local optimum for the CIP.

中文翻译：

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网络中断与有效阻隔下的共识

我们研究了共识动态下的网络鲁棒性问题。我们首先展示了共识拦截问题（CIP），其目标是在消除有限网络边缘的情况下最大化共识动态的收敛时间，可以被视为有效阻力拦截问题（ERIP）。然后，我们证明 ERIP 是强 NP 难的，即使对于直径为 3 且源/汇边缘固定的二部图也是如此。我们为 CIP 建立了相同的硬度结果，因此纠正了现有文献中的一些说法。然后我们证明 ERIP 和 CIP 都不允许多项式时间近似方案，而且，它们不能近似到（几乎）假设指数时间假设的多项式因子。最后，使用二次程序公式，我们为 ERIP 设计了一个多项式时间 $n^4$-近似算法，该算法仅取决于节点 $n$ 的数量并且与边缘电阻的大小无关。我们还开发了一种迭代启发式近似算法，以找到 CIP 的局部最优值。