Strongly Polynomial Parallel Work-Depth Tradeoffs for Directed SSSP
Adam Karczmarz, Wojciech Nadara, Marek Sokołowski
TL;DR
The paper resolves a longstanding question on strongly polynomial parallel SSSP for non-negative directed graphs by presenting deterministic work-depth tradeoffs that achieve sublinear depth while maintaining subquadratic, and in dense regimes near-linear, work. The core technical advance is a preprocessing framework that builds near-lists and permanently heavy vertices to enable efficient discovery steps, plus contractions that reuse computed structure across many steps. The results extend to dense graph regimes, produce immediate improvements for min-cost flow and assignment in parallel, and yield the first non-trivial dynamic algorithm for the minimum mean cycle, with extensions to lexicographic optimization and exponential weights in Word RAM models. Collectively, they push the boundary of parallel exact SSSP, unlock practical implications for related optimization problems, and open avenues for dynamic and weight-agnostic variants.
Abstract
In this paper, we show new strongly polynomial work-depth tradeoffs for computing single-source shortest paths (SSSP) in non-negatively weighted directed graphs in parallel. Most importantly, we prove that directed SSSP can be solved within $\tilde{O}(m+n^{2-ε})$ work and $\tilde{O}(n^{1-ε})$ depth for some positive $ε>0$. In particular, for dense graphs with non-negative real weights, we provide the first nearly work-efficient strongly polynomial algorithm with sublinear depth. Our result immediately yields improved strongly polynomial parallel algorithms for min-cost flow and the assignment problem. It also leads to the first non-trivial strongly polynomial dynamic algorithm for minimum mean cycle. Moreover, we develop efficient parallel algorithms in the Word RAM model for several variants of SSSP in graphs with exponentially large edge weights.