The Price of Feasibility: Greedy Approximation Bounds for String Supermodular Optimization over Oracle-Conditioned Greedoids

Greedy algorithms efficiently approximate combinatorial optimization problems, but their guarantees weaken when feasibility couples combinatorial structure with global physical constraints. We study monotone nondecreasing supermodular minimization over the bases of a graphic greedoid under physics-induced constraints. We model physics-informed selection using a look-ahead oracle that identifies candidates extendable to a feasible basis, yielding the Conditioned Sequential Greedy Algorithm. We derive a closed-form approximation bound, which we call the price of feasibility, based on the variability of oracle-restricted candidate sets and a probabilistic correction for unobserved elements. As a case study, we show that FORWARD, an algorithm for multi-source radial network reconfiguration, instantiates this framework. Numerical results demonstrate the tightness of the bound and quantify the feasibility-optimality trade-off.