Ride-hailing platforms increasingly rely on non-exclusive notifications--broadcasting a single request to multiple drivers simultaneously--to mitigate inefficiencies caused by uncertain driver acceptance. In this paper, the first in a two-part collaboration with Lyft, we formally model the Notification Set Selection Problem for a single decision cycle, where the platform determines the optimal subset of drivers to notify for each incoming ride request. We analyze this combinatorial optimization problem under two contention-resolution protocols: First Acceptance (FA), which prioritizes speed by assigning the ride to the first responder, and Best Acceptance (BA), which prioritizes match quality by selecting the highest-valued accepting driver. We show that welfare maximization under both mechanisms is NP-hard, ruling out a polynomial-time exact algorithm unless P=NP. Despite this hardness, we establish several positive algorithmic results. For FA, we develop a Polynomial Time Approximation Scheme (PTAS) for the single-rider problem. For the general matching setting, we provide a (1-1/e)-approximation algorithm that applies to both FA and BA. Along the way, we show that the FA valuation function constitutes a novel discrete choice model with structural properties of independent interest. For BA, we further prove that the welfare objective is monotone and submodular and exploit additional structure to obtain an approximation guarantee that surpasses the classical (1-1/e) barrier. Finally, when acceptance probabilities are independent of the drivers, we show that the BA problem can be solved exactly in polynomial time through a linear programming formulation. We evaluate our algorithms on synthetic instances and on instances calibrated using real ride-sharing data from Lyft, demonstrating strong empirical performance.

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