Best bounds for the longest path optimization problem in cubic Hamiltonian graph? optimization problem

Input: cubic Hamiltonian graph

feasible solution: A simple path

measure to optimize: length of the simple path

Design a polynomial-time algorithm that outputs the longest path it can find. 

To my knowledge, longest path problem does not have polynomial-time constant-factor approximation in cubic Hamiltonian graphs unless P=NP. This problem is interesting since the input is cubic graph and has a n-length Hamiltonian cycle.

On the Approximation of Finding A(nother) Hamiltonian Cycle in Cubic Hamiltonian Graphs, Journal of Algorithms 31, 249-268, 1999.
Check out
"Finding long paths and cycles in sparse Hamiltonian graphs"
by Feder, Motwani, and Subi from STOC 2000. AFAIK, this is the best algorithm known to date.