We propose a Branch-and-Cut algorithm for the robust influence maximization problem. The influence maximization problem aims to identify, in a social network, a set of given cardinality comprising actors that are able to influence the maximum number of other actors. We assume that the social network is given in the form of a graph with node thresholds to indicate the resistance of an actor to influence, and arc weights to represent the strength of the influence between two actors. In the robust version of the problem that we study, the node thresholds and arc weights are affected by uncertainty and we optimize over a worst-case scenario within given robustness budgets. We study properties of the robust solution and show that even computing the worst-case scenario for given robustness budgets is NP-hard. We implement an exact Branch-and-Cut as well as a heuristic Branch-Cut-and-Price. Numerical experiments show that we are able to solve to optimality instances of size comparable to other exact approaches in the literature for the non-robust problem, and we can tackle the robust version with similar performance. On larger instances ( $$\ge 2000$$ ≥ 2000 nodes), our heuristic Branch-Cut-and-Price significantly outperforms a 2-opt heuristic. An extended abstract of this paper appeared in the proceedings of IPCO 2019.

中文翻译：

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一种鲁棒影响最大化的精确算法

我们为鲁棒影响最大化问题提出了一种分支和切割算法。影响最大化问题旨在在社交网络中识别一组给定的基数，这些基数包括能够影响最大数量的其他参与者的参与者。我们假设社交网络以图表的形式给出，节点阈值表示一个参与者对影响的抵抗力，以及弧权重来表示两个参与者之间影响的强度。在我们研究的问题的稳健版本中，节点阈值和弧权重受不确定性的影响，我们在给定的稳健性预算内针对最坏情况进行优化。我们研究了鲁棒解决方案的特性，并表明即使计算给定鲁棒性预算的最坏情况也是 NP 难的。我们实现了一个精确的 Branch-and-Cut 和一个启发式的 Branch-Cut-and-Price。数值实验表明，对于非鲁棒性问题，我们能够解决与文献中其他精确方法相当的大小的最优实例，并且我们可以解决具有相似性能的鲁棒版本。在较大的实例（$$\ge 2000$$ ≥ 2000 个节点）上，我们的启发式 Branch-Cut-and-Price 显着优于 2-opt 启发式。本文的扩展摘要出现在 IPCO 2019 会议记录中。我们的启发式 Branch-Cut-and-Price 显着优于 2-opt 启发式。本文的扩展摘要出现在 IPCO 2019 会议记录中。我们的启发式 Branch-Cut-and-Price 显着优于 2-opt 启发式。本文的扩展摘要出现在 IPCO 2019 会议记录中。