Hamiltonian paths & social graph algorithm

Question

I have a random undirected social graph.

I want to find a Hamiltonian path if possible. Or if not possible (or not possible to know if possible in polynomial time) a series of paths. In this "series of paths" (where all N nodes are used exactly once), I want to minimize the number of paths and maximize the average length of the paths. (So no trivial solution of N paths of a single node).

I have generated an adjacency matrix for the nodes and edges already.

Any suggestions? Pointers in the right direction? I realize this will require heuristics because of the NP-complete (?) nature of the problem, and I am OK with a "good enough" answer. Also I would like to do this in Java.

Thanks!

Answer 1

Use a genetic algorithm (without crossover), where each individual is a permutation of the nodes. This gives you "series of paths" at each generation, evolving to a minimal number of paths (1) and a maximal avg. length (N).

Answer 2

If I'm interpreting your question correctly, what you're asking for is still NP-hard, since the best solution to the "multiple paths" problem would be a Hamiltonian path, and determining whether one exists is known to be NP-hard. Moreover, even if you're guaranteed that a Hamiltonian path doesn't exist, solving this problem could still be NP-hard, since I could give you a graph with a single disconnected node floating in space, for which the best solution is a trivial path containing that node and a Hamiltonian path in the remaining graph. As a result, unless P = NP, there isn't going to be a polynomial-time algorithm for your problem.

Hope this helps, and sorry for the negative result!

Answer 3

As you have realized there is no exact solution in polynomial time. You can try some random search methods though. My recommendation, start with genetic algorithm and try out tabu search.

Answer 4

Angluin and Valiant gave a near linear-time heuristic that works almost always in a sufficiently dense Erdos-Renyi random graph. It's described by Wilf, on page 121. Probably your random graph is not Erdos-Renyi, but the heuristic might work anyway (when it "fails", it still gives you a (hopefully) long path; greedily take this path and run A-V again).