On the Performance of a Simple Approximation Algorithm for the Longest Path Problem - Phuong, Nguyen Thi; Duc, Tran Vinh; Thanh, Le Cong (Journal of Computer Science and Cybernetics, 2019)
The longest path problem is known to be NP-hard. Moreover, they cannot be approximated within a constant ratio, unless ${\rm P=NP}$. The best known polynomial time approximation algorithms for this problem essentially find a path of length that is the logarithm of the optimum.In this paper we investigate the performance of an approximation algorithm for this problem in almost every case. We show that a simple algorithm, based on depth-first search, finds on almost every undirected graph $G=(V,E)$ a path of length more than $|V|-3\sqrt{|V| \log |V|}$ and so has performance ratio less than $1+4\sqrt{\log |V|/|V|}$.\