IP Models for Minimum Zero Forcing Sets, Forts, and Related Graph Parameters

2025-08-10

AI Summary

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.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Zero forcing is a binary coloring game on a graph where a set of filled vertices can force non-filled vertices to become filled following a color change rule. In 2008, the zero forcing number of a graph was shown to be an upper bound on its maximum nullity. In addition, the combinatorial optimization problem for the zero forcing number was shown to be NP-hard. Since then, the study of zero forcing and its related parameters has received considerable attention. In 2018, the forts of a graph were defined as non-empty subsets of vertices where no vertex outside the set has exactly one neighbor in the set. Forts have been used to model zero forcing as an integer program and provide lower bounds on the zero forcing number. To date, three integer programming models have been developed for the zero forcing number of a graph: the Infection Model, Time Step Model, and Fort Cover Model. In this article, we present variations of these models for computing the zero forcing number and related graph parameters, such as the minimum and maximum propagation times, throttling number, and fractional zero forcing number. In addition, we present several new models for computing the realized propagation time interval, all minimal forts of a graph, and the fort number of a graph. We conclude with several numerical experiments that demonstrate the effectiveness of our models when applied to small and medium order graphs. Moreover, we provide experimental evidence for several open conjectures regarding the propagation time interval, the number of minimal forts, the fort number, and the fractional zero forcing number of a graph.

Problem

• Develop IP models for zero forcing number computation
• Explore forts and related graph parameters optimization
• Validate models via numerical experiments and conjectures

Innovation

• Integer programming models for zero forcing
• New models for propagation time intervals
• Numerical experiments on graph parameters