Finding the hamiltonian path in a directed cyclic graph

Question

i want to know if there is an algorithm to find the longest cyclic path in a directed weighted graph (i think this i a problem of finding the largest Hamiltonian sub-graph).

I need to start from one vertex and return to the same vertex, whit the most possible nodes traversed.

Thanks

Do you want the most possible nodes, or the highest path weight? — bdonlan, Mar 27 '11 at 00:47
Sorry, i would like the most possible nodes, although i am interested in a solution with the highest path weight — Joksim, Mar 27 '11 at 00:59

score 1 · Answer 1 · answered Mar 27 '11 at 04:37

This problem is a special case of the optimal euler circuit problem where all edge weights are 1; the original problem is NP-complete. Moreover, this problem can be used to solve the Hamiltonian Graph problem (a Hamiltonian cycle exists if and only if the optimal circuit traverses all nodes), so it remains NP-complete even with the special case restriction. Any exact solution will (unless P = NP) require exponential time. You may find this paper helpful; it describes a polynomial-time approximation algorithm for this problem, as well as a polynomial-time algorithm for cases where the graph has at most degree 4:

Qiao, Yu. "Optimal Euler Circuit of Maximum Contiguous Cost." IEICE Trans. Fundamentals E90-A, no. 1 (2007): 274-280.

score 0 · Answer 2 · answered Mar 27 '11 at 05:24

0

A good approximation gives the Hilbert Curve.

answered Mar 27 '11 at 05:24

Micromega

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Finding the hamiltonian path in a directed cyclic graph

2 Answers2