A greedy approximation algorithm for the longest path problem in undirected graphs

TL;DR

This paper introduces a greedy algorithm with quadratic time complexity for approximating the longest path in large undirected graphs, addressing the NP-hardness of the problem.

Contribution

It presents a new greedy approximation algorithm for the longest path problem in undirected graphs with a specific polynomial time complexity.

Findings

The algorithm runs in O(n^2 (n+m)) time.

It provides a quick approximation method for large graphs.

The approach is practical for large-scale graph analysis.

Abstract

In graph theory, the longest path problem is the problem of finding a simple path of maximum length in a given graph. For some small classes of graphs, the problem can be solved in polynomial time [2, 4], but it remains NP-hard on general graphs, since it includes the Hamiltonian path problem as a special case [3]. Motivated by finding a simple, quick algorithm for finding long paths in large graphs, in this paper we show a greedy algorithm with a time complexity of O(n^2 (n+m)), where n is the number of the vertices and m is the number of edges.