Does anyone have any idea about how to find all paths from a source vertex to a target vertex under a specific weight in a graph?
PS, the graph is huge in my case (around million vertices), but a general efficient algorithm would also be appreciated.
Please don't say to run Yen's algorithm with some search for the "correct" k.
Thanks.
There is an answer on mathoverflow for finding all paths in an undirected graph. You can modify this answer to include weight consideration. As he stated in his answer
The idea is very simple: Do an exhaustive search, but bail early if you've gotten yourself into a corner.
In addition to that, you will have to bail when the weight is off the limit.
Related
So I have the following layout:
graph representation
The objective is to collect all the yellow blocks by moving the white ball around. I'm trying to come up with an algorithm that will calculate an efficient path however I'm not too sure where to start.
Initially I thought about path finding algorithms like Djikstra and A* but they don't seem to fit with my goal. I've also thought about hamiltonian paths which is closer to what I want but still doesn't seem to solve the problem.
Any suggestions on what sort of algorithm can be used would be appreciated.
Your problem has a classic name in the litterature, it is the minimum hamiltonian walk problem. Beware not to mistake it with the minimum hamiltonian path problem, its 'cousin', because it is much more famous, and much, much harder (finding a hamiltonian walk can be done in polynomial time, finding a hamiltonian path is NP-complete). The traveling salesman problem is the other name of the minimum hamiltonian path problem (path, not walk).
There are very few ressources on this problem, but nevertheless you can have a look at an article called 'An algorithm for finding a short closed spanning walk in a graph' by Takamizawa, Nishizeki and Saito from 1980. They provide a polynomial algorithm to find such a path.
If the paper is a bit hard to read, or the algorithm too complex to implement, then I'll suggest that you go for the christofides algorithm, because it runs in polynomial time, and is somehow efficient (it is a 2-approximation if I remember well).
Another possible approach would be to go for a greedy algorithm, like a nearest unvisited neigbor algorithm (start somewhere, go to the nearest node that is not in the walk yet, repeat until everyone is in the walk).
Acutally, I think the easiest and maybe best simple solution is to go greedy.
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!
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!
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).
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).
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.
I need to know if it is possible to find a simple path with maximum cost in any weighted undirected graph.
I mean to find THE MOST expensive path of all for any pair of vertex.
Input: Graph G = (V,E)
Output: The cost of the most expensive path in the graph G.
Is this problem NP-Complete?, I think it is. Could you provide any reference to an article where I can review this.
You're not the first to think of this problem. In fact, it was the first link in the google search results.
edit
Guys, un-weighted graph is a special case of weighted graph: all edges have weight 1 :)
This is similar to traveling salesman, except your heuristic is the Max and not Min. Read up on the traveling salesman.
The problem is NP complete because it can be derived from a problem that is already proven to be NP-Complete (Traveling salesman). The answer is checkable in polynomial time, but an answer cannot be found in polynomial time.
Read http://en.wikipedia.org/wiki/Travelling_salesman_problem
Yes, this problem is NP because you are asking for the maximum which means that you'll need to go through all possible paths. The decision version of this problem ("is there a path of length n?") is known NP-complete (as noted above).
What are the efficient algorithms for finding a vertex tour in a weighted undirected graph with maximum cost if we need to start from a particular vertex?
It's NPC because if you set weights as 1 for all edges, if HC exists it will be your answer, and so In all you can find HC existence from a single source which is NPC by solving this problem so your problem is NPC, but there are some polynomial approximation algorithms.
Since the problem is NP-hard, you are very unlikely to find an efficient algorithm that solves the problem exactly for all possible weighted input graphs.
However, there might be efficient algorithms that are guaranteed to find an answer that is at most a constant times away from the best possible answer, e.g. there might be an efficient algorithm that is guaranteed to find a path that has weight at least 1/2 of the maximum weight path.
If you are interested in searching for such algorithms, you could try Google searches for "weighted hamiltonian path approximation algorithm", which is close to, but not identical to, your problem. It is not the same because Hamiltonian paths are required to include all vertexes. Here is one research paper that might either contain, or have ideas that lead to, an approximation algorithm for your problem:
http://portal.acm.org/citation.cfm?id=139404.139468
"A general approximation technique for constrained forest problems" by Michel X. Goemans and David P. Williams.
Of course, if your graphs are small enough that you can enumerate all possible paths containing your desired vertex "fast enough for your purposes", then you can solve it exactly.
I have reduced my problem to finding the minimal spanning tree in the graph. But I want to have one more constraint which is that the total degree for each vertex shouldnt exceed a certain constant factor. How do I model my problem? Is MST the wrong path? Do you know any algorithms that will help me?
One more problem: My graph has duplicate edge weights so is there a way to count the number of unique MSTs? Are there algorithms that do this?
Thank You.
Edit: By degree, I mean the total number of edges connecting the vertex. By duplicate edge weight I mean that two edges have the same weight.
Well, it's easy to prove that there may not be a solution: just make your input graph a tree that has a vertex with degree higher than your limit..
Garey Johnson had this problem reduce to hamilton :( So this one helped. Approximating the first one: http://caislab.icu.ac.kr/Lecture/data/2003/spring/ice514/project/m03.ppt
However, better working models are appreciated...
Counting: http://mathworld.wolfram.com/SpanningTree.html . According to this, mathematica has a function. Any suggestions in this one?
This paper shows how to find, in polynomial time, a spanning tree of maximum degree d + 1 that is at least as good as any spanning tree of maximum degree d: http://www.andrew.cmu.edu/user/mohits/mbdst.pdf
//Edit The original link is currently inactive, try http://research.microsoft.com/pubs/80193/mbdst.pdf instead.