IP Models for Minimum Zero Forcing Sets, Forts, and Related Graph Parameters
Computing exact values of zero-forcing-related parameters in graph theory—such as minimum zero-forcing set, propagation time (min/max), throttling number, fractional zero-forcing number, fortress number, and all minimal fortress sets—remains computationally challenging due to their NP-hardness. Method: We develop novel integer programming (IP) formulations, introducing three modeling paradigms: infection dynamics, temporal evolution, and coverage constraints. Our models enable the first complete enumeration of propagation time intervals and systematic enumeration of all minimal fortress sets. Results: Evaluated on small-to-medium graphs, the IP models efficiently solve multiple NP-hard zero-forcing parameters. They yield the first large-scale numerical evidence for long-standing open conjectures—including bounds on throttling numbers and relationships between fortress and zero-forcing numbers—thereby advancing zero-forcing theory from existential analysis toward computational tractability and empirical validation.