Construct a minimum spanning tree covering a specific subset of the vertices

Question

I have an undirected, positive-edge-weight graph (V,E) for which I want a minimum spanning tree covering a subset k of vertices V (the Steiner tree problem).

I'm not limiting the size of the spanning tree to k vertices; rather I know exactly which k vertices must be included in the MST.

Starting from the entire MST I could pare down edges/nodes until I get the smallest MST that contains all k.

I can use Prim's algorithm to get the entire MST, and start deleting edges/nodes while the MST of subset k is not destroyed; alternatively I can use Floyd-Warshall to get all-pairs shortest paths and somehow union the paths. Are there better ways to approach this?

Solution 1

There's a lot of confusion going on here. Based on what the OP says:

I'm not limiting the size of the spanning tree to k vertices; rather I know exactly which k vertices must be included in the MST.

This is the Steiner tree problem on graphs. This is not the k-MST problem. The Steiner tree problem is defined as such:

Given a weighted graph G = (V, E), a subset S ⊆ V of the vertices, and a root r ∈ V , we want to find a minimum weight tree which connects all the vertices in S to r. 1

As others have mentionned, this problem is NP-hard. Therefore, you can use an approximation algorithm.

Early/Simple Approximation Algorithms

Two famous methods are Takahashi's method and Kruskal's method (both of which have been extended/improved by Rayward-Smith):

• Takahashi H, Matsuyama A: An approximate solution for the Steiner problem in graphs. Math. Jap 1980, 24:573–577.
• Kruskal JB: On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem. In Proceedings of the American Mathematical Society, Volume 7. ; 1956:48–50.
• Rayward-Smith VJ, Clare A: On finding Steiner vertices. Networks 1986, 16:283–294.

---

Shortest path approximation by Takahashi (with modification by Rayward-Smith)

---

Kruskal's approximation algorithm (with modification by Rayward-Smith)

---

Modern/More Advanced Approximation Algorithms

In biology, more recent approaches have treated the problem using the cavity method, which has led to a "modified belief propagation" method that has shown good accuracy on large data sets:

• Bayati, M., Borgs, C., Braunstein, A., Chayes, J., Ramezanpour, A., Zecchina, R.: Statistical mechanics of steiner trees. Phys. Rev. Lett. 101(3), 037208 (2008) 15.
• For an application: Steiner tree methods for optimal sub-network identification: an empirical study. BMC Bioinformatics. BMC Bioinformatics 2013 30;14:144. Epub 2013 Apr 30.

In the context of search engine problems, approaches have focused on efficiency for very large data sets that can be pre-processed to some degree.

• G. Bhalotia, A. Hulgeri, C. Nakhe, S. Chakrabarti, and S. Sudarshan. Keyword Searching and Browsing in Databases using BANKS. In ICDE, pages 431–440.
• G. Kasneci, M. Ramanath, M. Sozio, F. M. Suchanek, and G. Weikum. STAR: Steiner-tree approximation in relationship graphs. In ICDE’09, pages 868–879, 2009

Solution 2

The problem you stated is a famous NP-hard problem, called Steiner tree in graphs. There are no known solutions in polynomial time and many believe no such solutions exist.

Solution 3

Run Prim's algorithm on the restricted graph (k, E') where E' = {(x, y) ∈ V : x ∈ k and y ∈ k}). Constructing that graph takes O(|E|).

Comments

[Mat]

Not sure I understand, but can't you just run your favorite MST algo on (k,E)?

[Savino Sguera]

Uhm, how is this different from removing the unwanted vertices and running Prim (or Kruskal) on the remaining ones?

[sehe]

i'd be thinking 'subgraphs' there

[atp]

If I remove the unwanted vertices I might also lose intermediate edges that connect k vertices that are far apart. For example if I have: k--o--o--o--k where o represents an unnecessary vertex and k represents one I need, if I deleted the middle o there would be no way to construct the MST between my k vertices.

[aioobe]

So you interested in the minimum spanning tree, which doesn't necessarily span all vertices, only the vertices in k?

[atp]

Exactly. The MST that includes all of k at least, and then as little else as possible.

[Fred Foo]

@Jasie: then you can't get a minimum spanning tree because the subgraph is not connected.

[phoenix]

Hi could you solve your problem? If possible can you help with the pseudo code/code? I have similar problem but the graph is unweighted.

[Palec]

The question is unclear about whether k is a number or a set. Will you please clarify?

[aioobe]

@ash, would you mind un-accepting my answer? As pointed out in the comments, the algorithm is faulty, and I'd like to delete it to avoid spreading misinformation.

[atp]

@aioobe done -- I haven't looked at this in a while but do you know if any of the other answers are correct?

[aioobe]

Nope, unfortunately not. :-/

[Timothy Dalton]

You can solve this problem using the Steiner Graph implementation given in Networkx - networkx.github.io/documentation/stable/reference/algorithms‌​/… steiner_tree(G, terminal_nodes, weight='weight') and some examples can be found in this thread gis.stackexchange.com/questions/307336/…

[Ankit Roy]

This might work alright some of the time, but it's not even guaranteed that the E' is connected -- and even if it is, it might be possible to save arbitrarily much distance by introducing a Steiner point (i.e., a vertex not in k). (Less than "arbitrarily much" if the distances obey the Triangle Inequality, but nothing says they have to.)

[user2398029]

@j_random_hacker interested in posting an alternative solution?

[Ankit Roy]

@user2398029: I upvoted meh's answer (and I don't know why "Bill the Lizard" deleted adi's much earlier answer saying mostly the same thing). Basically this is an NP-hard problem to solve optimally; if you google "Steiner tree approximation" you can probably get some OK algorithms.

[Ankit Roy]

@user2398029: It might be helpful to look at chapter 3 of this link from adi's answer: cc.gatech.edu/fac/Vijay.Vazirani/book.pdf. (I (re)post this here since I can see deleted posts, but I'm not sure what the rep cutoff is for that.)

[Palec]

Actually, Steiner tree in graphs has a fixed set of k vertices as input, while the OP gives just the k and lets the algorithm find the set. This problem is called k-MST and is also NP-hard. See also problems related to MST on Wikipedia.

[user2398029]

@Palec Actually, that is wrong. "I'm not limiting the size of the spanning tree to k vertices; rather I know exactly which k vertices must be included in the MST." This problem is the Steiner tree problem.

[user2398029]

Also, -1 to @meh because the fact that the problem is NP-hard doesn't mean we can't get useful solutions with approximation algorithms. This answer does not help the OP in solving his problem.

[Jeff Bezos]

Thank you so much for this. This post led me to a nice R implementation in the SteinerNet package