在离散动力系统中寻找非平凡最小不动点
Finding Nontrivial Minimum Fixed Points in Discrete Dynamical Systems
AI总结：
本文研究离散动力系统中受影响节点最少的非平凡不动点问题，证明其近似难度，并提出整数线性规划、启发式框架和贪心方法以高效求解。
摘要：
网络化离散动力系统常被用于建模传染病的传播以及协调博弈中智能体的决策行为。此类动力系统的不动点表示系统收敛后所达到的配置。在传播不良传染（如谣言和错误信息）时，收敛到受影响节点数量较少的不动点是一个理想目标。基于这些考虑，我们形式化了一个新的优化问题：寻找系统中受影响节点数量最少的非平凡不动点。我们证明，除非 P = NP，否则对于任意常数 epsilon > 0，不存在多项式时间算法能够将该问题的解近似到 n^1-ε 的因子以内。为了应对这种计算难解性，我们识别出若干可以在多项式时间内高效求解的特殊情形。此外，我们引入了一个整数线性规划，用于处理规模适中的网络上的该问题。为了在更大规模的网络上求解该问题，我们提出了一个通用的启发式框架以及贪心选择方法。在真实世界网络上的大量实验结果表明了所提启发式方法的有效性。
英文摘要：
Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the dissemination of undesirable contagions (such as rumors and misinformation), convergence to fixed points with a small number of affected nodes is a desirable goal. Motivated by such considerations, we formulate a novel optimization problem of finding a nontrivial fixed point of the system with the minimum number of affected nodes. We establish that, unless P = NP, there is no polynomial time algorithm for approximating a solution to this problem to within the factor n^1-εfor any constant epsilon > 0. To cope with this computational intractability, we identify several special cases for which the problem can be solved efficiently. Further, we introduce an integer linear program to address the problem for networks of reasonable sizes. For solving the problem on larger networks, we propose a general heuristic framework along with greedy selection methods. Extensive experimental results on real-world networks demonstrate the effectiveness of the proposed heuristics.
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