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### Largest cycle in the graph

by artist (Parson)
 on Aug 25, 2003 at 18:48 UTC Need Help??

artist has asked for the wisdom of the Perl Monks concerning the following question:

Dear Monks,

This is an algorithm question, but I think that there should be a Perl implementation available. How I can find the Largest path and cycle in the given graph. (Without visiting any node twice). Largest= Maximum number of nodes.

Thanks
artist

Replies are listed 'Best First'.
Re: Largest cycle in the graph
by liz (Monsignor) on Aug 25, 2003 at 19:28 UTC
Re: Largest cycle in the graph
by blokhead (Monsignor) on Aug 25, 2003 at 19:10 UTC
The longest cycle problem includes the special case of finding a Hamiltonian cycle (a cycle that visits every node exactly once), and is therefore NP-complete. This means you will not find an efficient solution, only something along the lines of a brute-forcing. I couldn't find much info on the longest-cycle problem, but you can reduce the longest-path problem to the Hamiltonian cycle problem as follows:
If you have a graph with V vertices, take all size-V subsets of vertices, then all size-(V-1) subsets, etc.. For each of these subsets of vertices, determine if there's a Hamiltonian cycle. The first subset of nodes you encounter that contains a Hamiltonian cycle is the longest cycle in your original graph.
The Hamiltonian cycle problem is very common, and you should be able to find a lot of literature on the subject. However, any code is going to be exponential (with the possible exception of some special class of graphs perhaps).

It may be time to rethink a way through this problem. Is there is a way to do it without needing the longest cycle? This page has some good suggestions for rethinking problems involving Hamiltonian cycles. For instance, perhaps you can solve the same problem only by needing an Eulerian cycle, which is much easier to compute.

First off, I know this response is not going to solve your problem. I do not know of an implementation you can use immediately. I do know that Graph::Base has some useful methods that you could use in writing your own solution. For example, methods are available to return biconnectedness and mean density statistics.

The only information (I'm no expert on graph theory!) that I can add to the NP-completeness issue comes from Steven S. Skiena's excellent The Algorithm Design Manual, page 324, which provides a thorough overview of this class of problem, including references to implementations.

"For sufficiently dense graphs, there always exists at least one Hamiltonian cycle, and further, such a cycle can be found quickly. An efficient algorithm for finding a Hamiltonian cycle in a graph where all vertices have degree >= n/2 is given in U. Manber. Introduction to Algorithms."

Here's hoping that your graphs are sufficiently dense! Happy solving.

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