Finding paths of length k in O*(2^k) time (0807.3026v3)

Abstract: We give a randomized algorithm that determines if a given graph has a simple path of length at least k in O(2^k poly(n,k)) time.

• The paper introduces a randomized algorithm that efficiently finds simple paths of length k in a graph, achieving a significantly improved runtime of O*(2^k).
• The algorithm leverages algebraic tools within the group algebra GF(2)[Z] and a probabilistic approach based on the Schwartz-Zippel Lemma to achieve its efficiency.
• This algorithmic advancement offers substantial computational improvements for NP-hard pathfinding problems and suggests potential applications to related areas like Hamiltonian paths.

An Efficient Algorithm for Detecting k-Path in Graphs

This paper presents a notable advancement in addressing the k-path problem, a fundamental challenge in graph theory and computer science, known for its NP-completeness. The contribution is a randomized algorithm capable of determining whether a given graph contains a simple path of length at least k, achieving a running time of O*(2^k), where the notation O* suppresses polynomial factors in n and k.

Context and Development

The problem of finding a simple path of length k, or a k-path, in a graph is central to a variety of practical and theoretical applications. Traditional approaches yield computational inefficiencies, given that the naive algorithm operates with a time complexity of O(n^k), which is only feasible for relatively small values of k. Previous attempts to reduce this dependency included Monien's algorithms running in O*(k!) time and Alon, Yuster, and Zwick's algorithms achieving O*((2e)^k) and O*(c^k) time, where c is a significantly larger constant.

More recent endeavors resulted in algorithms running in O*(4^k) time. Koutis further improved the runtime to O*(2.83^k) by proposing a method for identifying k-subgraphs with an odd number of k-paths in O*(2^k) time. Building on Koutis' methodologies, this paper successfully extends these results to detect and produce k-paths in O*(2^k) time, matching the known complexity for solving the problem when k equals n, the number of graph nodes.

Algorithmic Approach

The algorithm relies heavily on algebraic tools, specifically working within the group algebra GF(2)[Z] for efficient polynomial identity testing. This approach simplifies the process by substituting algebraic elements for path variables to isolate and evaluate multilinear terms that correspond to k-paths.

Key to the algorithm's effectiveness is its probabilistic structure, leveraging the Schwartz-Zippel Lemma to maintain high accuracy in determining the presence of a k-path. By using random selections of vector and field elements, the algorithm reduces multilinear polynomial evaluations effectively. The correctness of the approach hinges on the fact that if a multilinear term exists, then its presence is confirmed with a probability of at least 1/5, sufficient when repeated over multiple trials.

Implications and Open Questions

This development presents significant computational improvements for tackling path-related NP-hard problems, with potential extensions to related domains such as Hamiltonian paths and tours, where path length constraints are critical. Moreover, the techniques employed may have broader implications for probabilistic evaluation methods in combinatorial optimization.

The paper identifies several open questions, applying particular interest in extending the algorithm to weighted graphs through the "SHORT CHEAP TOUR" problem, and exploring deterministic counterparts that could achieve similar time complexities. Resolving these questions could further integrate this algorithmic mindset into broader graph processing tasks.

Conclusion

In summary, the paper contributes a robust method for addressing the k-path problem, reducing the computational overhead substantially through innovative use of algebraic principles and randomized algorithms. This accomplishment offers new avenues for resolving classical graph-related challenges and invites further exploration into refining these techniques for greater applicability in weighted and deterministic contexts.