On Survival Time in Robust Optimization Over Time for Discrete Search Spaces

Robust Optimization Over Time (ROOT) addresses dynamic optimization problems where the goal is not to identify the best solution at each instant, but to find solutions that remain effective across multiple future environments. Despite recent algorithmic advances, the theoretical foundations of ROOT remain limited, especially regarding its formulation in terms of survival time. Additionally, discrete problem settings have received little attention. This paper addresses these limitations from a theoretical perspective by analyzing survival time in discrete ROOT under stochastic k-bit-flip environmental dynamics. Using drift analysis and martingale techniques, we derive tight analytical bounds and identify phase transitions for survival time on dynamic OneMax, capturing gradual fitness degradation. For dynamic LeadingOnes, which models catastrophic structural failure, we prove that survival time follows an exact geometric distribution. Monte Carlo simulations closely match all theoretical predictions. The results connect ROOT with classical runtime analysis and formalize the mechanisms governing solution durability, providing a mathematical foundation for predictive robustness models, survival-aware heuristics, and deployment policies tailored to combinatorial environments.