Strongly Polynomial Parallel Work-Depth Tradeoffs for Directed SSSP

2025-10-22

AI Summary

This paper addresses the single-source shortest paths (SSSP) problem on nonnegative weighted directed graphs. It presents the first strongly polynomial, approximately work-efficient, and sublinear-depth parallel algorithm for SSSP. The method introduces novel parallel techniques that operate in the Word RAM model and handle exponentially large edge weights, thereby breaking the work–depth trade-off barrier inherent in dense graphs. Key contributions include: (1) an algorithm achieving $ ilde{O}(m + n^{2-epsilon})$ work and $ ilde{O}(n^{1-epsilon})$ depth—significantly improving upon prior strongly polynomial parallel SSSP algorithms; (2) the first approximately work-efficient, strongly polynomial parallel SSSP algorithm for nonnegative-weight dense graphs; and (3) the first nontrivial strongly polynomial dynamic algorithm for the minimum mean cycle problem. These advances enhance the parallel efficiency of fundamental combinatorial optimization problems, including minimum-cost flow and assignment.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $ ilde{O}(m+n^{2-epsilon})$ work and $ ilde{O}(n^{1-epsilon})$ depth for some positive $epsilon>0$. 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.

Problem

• Develop work-depth tradeoffs for parallel directed SSSP
• Achieve sublinear depth for dense weighted graphs
• Enable improved strongly polynomial parallel flow algorithms

Innovation

• Strongly polynomial work-depth tradeoffs for SSSP
• Sublinear depth algorithm for dense graphs
• Parallel algorithms for large edge weights