In the Serial Parallel Decision Problem (SPDP), introduced by Kuszmaul and Westover [SPAA'24], an algorithm receives a series of tasks online, and must choose for each between a serial implementation and a parallelizable (but less efficient) implementation. Kuszmaul and Westover describe three decision models: (1) Instantly-committing schedulers must decide on arrival, irrevocably, which implementation of the task to run. (2) Eventually-committing schedulers can delay their decision beyond a task’s arrival time, but cannot revoke their decision once made. (3) Never-committing schedulers are always free to abandon their progress on the task and start over using a different implementation. Kuszmaul and Westover gave a simple instantly-committing scheduler whose total completion time is 3-competitive with the offline optimal schedule, and proved two lower bounds: no eventually-committing scheduler can have competitive ratio better than ϕ ≈ 1.618 in general, and no instantly-committing scheduler can have competitive ratio better than 2 in general. They conjectured that the three decision models should admit different competitive ratios, but left upper bounds below 3 in any model as an open problem. In this paper, we show that the powers of instantly, eventually, and never committing schedulers are distinct, at least in the "massively parallel regime". The massively parallel regime of the SPDP is the special case where the number of available processors is asymptotically larger than the number of tasks to process, meaning that the work associated with running a task in serial is negligible compared to its runtime. In this regime, we show (1) The optimal competitive ratio for instantly-committing schedulers is 2, (2) The optimal competitive ratio for eventually-committing schedulers lies in [1.618, 1.678], (3) The optimal competitive ratio for never-committing schedulers lies in [1.366, 1.500]. We additionally show that our instantly-committing scheduler is also 2-competitive outside of the massively parallel regime, giving proof-of-concept that results in the massively parallel regime can be translated to hold with fewer processors.

In sparse Erdős-Rényi graphs, it is known that a linear-time algorithm of Karp and Sipser achieves near-optimal matching sizes asymptotically almost surely, giving a law-of-large numbers for the matching numbers of such graphs in terms of solutions to an ODE [Jonathan Aronson et al., 1998]. We provide an extension of this analysis, identifying broad ranges of stochastic block model parameters for which the Karp-Sipser algorithm achieves near-optimal matching sizes, but demonstrating that it cannot perform optimally on general stochastic block model instances. We also consider the problem of constructing a matching online, in which the vertices of one half of a bipartite stochastic block model arrive one-at-a-time, and must be matched as they arrive. We show that, when the expected degrees in all communities are equal, the competitive ratio lower bound of 0.837 found by Mastin and Jaillet for the Erdős-Rényi case [Andrew Mastin and Patrick Jaillet, 2013] is achieved by a simple greedy algorithm, and this competitive ratio is optimal. We then propose and analyze a linear-time online matching algorithm with better performance in general stochastic block models.