Negative-Weight Single-Source Shortest Paths in Near-linear Time (2203.03456v5)

Abstract: We present a randomized algorithm that computes single-source shortest paths (SSSP) in $O(m\log^8(n)\log W)$ time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $\tilde O((m+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] and $m^{4/3+o(1)}\log W$ [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic $\tilde O(m\sqrt{n}\log W)$ bound from over three decades ago [Gabow and Tarjan SICOMP'89].

• The paper introduces a randomized combinatorial algorithm that computes single-source shortest paths on negative-weight graphs in O(m log^8(n) log W) time.
• It employs low-diameter decomposition and recursive scaling to simplify the problem, avoiding complex continuous optimization techniques.
• This breakthrough outperforms the classic O(m√n log W) bound, offering practical benefits for resource-constrained and real-world applications.

An Algorithm for Single-Source Shortest Paths with Negative Weights in Near-Linear Time

The research paper presents a sophisticated yet elegantly simple randomized algorithm that calculates single-source shortest paths (SSSP) in graphs with negative weights in near-linear time, specifically $O(m\log^8(n)\log W)$, where $m$ and $n$ represent the number of edges and vertices, respectively, and $W$ is a parameter related to the minimal edge weight. This complexity essentially addresses and resolves a longstanding problem in graph algorithms efficiently. Prior to this work, state-of-the-art solutions such as those presented in FOCS'20 tackled this problem within complexity bounds of $\tilde O((m+n^{1.5})\log W)$ and $m^{4/3+o(1)}\log W$ for dense and sparse graphs.

The significant contribution of this paper is its simplicity. Unlike contemporary methods that rely heavily on intricate continuous optimization and dynamic algorithms, this work introduces a novel combinatorial approach. Notably, this is the first algorithm to surpass the $O(m\sqrt{n}\log W)$ bound—an achievement that has remained unchallenged for over thirty years since established by Gabow and Tarjan.

Key Components of the Algorithm

1. Low-Diameter Decomposition: The algorithm employs a decomposition technique where the graph is broken into subgraphs of small diameter. This decomposition is instrumental for efficiently handling subgraphs by ensuring that each strongly connected component (SCC) has a bounded diameter. The decomposition considers only non-negative-weight edges, a prominent feature distinguishing it from earlier methodologies.
2. Recursive Scaling: The core of the algorithm leverages a recursive scaling technique. By iteratively refining edge weight approximations and reducing the search space, the algorithm progressively approaches the shortest path solution without tripping the penalty of negative weight cycles.
3. Combinatorial Approach: The implementation primarily uses basic combinatorial structures and tools instead of advanced algebraic or flow algorithms, making it more accessible and potentially more efficient in various computational environments.

Theoretical and Practical Implications

The results have concrete implications for both theoretical exploration and practical application in computer science:

• Theoretical:

The method breaks the myth surrounding the necessity of complex optimization methods for negative-weight graphs. By challenging assumptions about the difficulty of the problem, it opens avenues for further exploration into simple combinatorial algorithms for other graph problems. It invites reconsideration of existing graph algorithm complexities and potential simplifications.

• Practical:

From a practical standpoint, the simplicity paired with the near-linear efficiency of the proposed algorithm makes it an attractive candidate for integration into software requiring SSSP calculations on negatively weighted graphs. Having a method that does not depend on complex mechanistic frameworks means improved maintainability and usability in resource-constrained environments.

Speculations on Future Development

This work potentially paves the way for further simplification and optimization of algorithms within the domain of graph theory and network flows. Future studies can expand upon:

1. Generalization to Other Problems: The insights and methodologies gleaned from this paper can be adapted and extended to tackle other graph problems like transshipment and min-cost flow, optimizing them similarly via combinatorial means.
2. Parallelization and Distributed Computing: Implementing the presented concepts in parallel or distributed computing environments could yield performance improvements commensurate with multi-core or distributed systems.
3. Optimizing Polylogarithmic Factors: While the authors acknowledge that logarithmic factors in the runtime are not optimized, future research can focus on refining these to improve the practical performance further.

In conclusion, this paper introduces a promising advance in the field of graph algorithms, making complex SSSP problems with negative weights more tractable via a noteworthy combinatorial, and simple, framework.