A branch-and-cut approach for the least cost influence problem on social networks

This paper studies a problem in the online targeted marketing setting called the least cost influence problem (LCIP) that is known to be NP-hard. The goal is to find the minimum total amount of inducements (individuals to target and associated tailored incentives) required to influence a given population. We develop a branch-and-cut approach to solve this LCIP on arbitrary graphs. We build upon Gunnec et al.'s novel totally unimodular (TU) formulation for the LCIP on trees. The key observation in applying this TU formulation to arbitrary graphs is to enforce an exponential set of inequalities that ensure the influence propagation network is acyclic. We also design several enhancements to the branch-and-cut procedure that improve its performance. We provide a large set of computational experiments on real-world graphs with up to 155 000 nodes and 327 000 edges that demonstrates the efficacy of the branch-and-cut approach. This branch-and-cut approach finds solutions that are on average 1.87% away from optimality based on a test-bed of 160 real-world graph instances. We also develop a heuristic that prioritizes nodes that receive low influence from their peers. This heuristic works particularly well on arbitrary graphs, providing solutions that are on average 1.99% away from optimality. Finally, we observe that partial incentives can result in significant cost savings, over 55% on average, compared to the setting where partial incentives are not allowed.