There has been significant progress in understanding the parallelism inherent to iterative sequential algorithms: for many classic algorithms, the depth of the dependence structure is now well understood, and scheduling techniques have been developed to exploit this shallow dependence structure for efficient parallel implementations. A related, applied research strand has studied methods by which certain iterative task-based algorithms can be efficiently parallelized via relaxed concurrent priority schedulers. These allow for high concurrency when inserting and removing tasks, at the cost of executing superfluous work due to the relaxed semantics of the scheduler.In this work, we take a step towards unifying these two research directions, by showing that there exists a family of relaxed priority schedulers that can efficiently and deterministically execute classic iterative algorithms such as greedy maximal independent set (MIS) and matching. Our primary result shows that, given a randomized scheduler with an expected relaxation factor of k in terms of the maximum allowed priority inversions on a task, and any graph on n vertices, the scheduler is able to execute greedy MIS with only an additive factor of poly(k) expected additional iterations compared to an exact (but not scalable) scheduler. This counter-intuitive result demonstrates that the overhead of relaxation when computing MIS is not dependent on the input size or structure of the input graph. Experimental results show that this overhead can be clearly offset by the gain in performance due to the highly scalable scheduler. In sum, we present an efficient method to deterministically parallelize iterative sequential algorithms, with provable runtime guarantees in terms of the number of executed tasks to completion. arXiv:1808.04155v1 [cs.DS] 13 Aug 2018 may only depend on a small subset of other nodes, namely its neighbors which are higher priority in the random order. Blelloch, Fineman and Shun [7] performed an in-depth study of the asymptotic properties of this dependence structure, proving that, for any graph, the maximal depth of a chain of dependences is in fact O(log 2 n) with high probability, where n is the number of nodes in the graph. Recently, an impressive analytic result by Fischer and Noever [13] provided tight Θ(log n) bounds on the maximal dependency depth for greedy sequential MIS, effectively closing this problem for MIS. Beyond greedy MIS, there has been significant progress in analyzing the dependency structure of other fundamental sequential algorithms, such as algorithms for matching [7], list contraction [25], Knuth shuffle [25], linear programming [5], and graph connectivity [5].An alternative approach has been to employ relaxed data structures to schedule task-based programs. Starting with Karp and Zhang [19], the general idea is that, in some applications, the scheduler can relax the strict order induced by following the sequential algorithm, and allow tasks to be processed speculatively ahead of their dependencies, without loss o...