I have to create a solution for a weighted undirected graph, passing through all the nodes, with a total minimum cost. Several paths, with no defined starting nodes, should end up and meet at one intersecting node. The number of the paths, and the number of the nodes included in a path are not pre-determined. The nodes can be passed more than once.
What kind of problem am I dealing with, possible algorithms as solution?
I suppose it should be a variation of a Minimum spanning tree (meaning using the intersection node as a starting point for the paths in stead of ending point)
It's called Minimum Cost Hamiltonian Circuit problem.
Here you can read more about it.
It is a tree you are looking for and the problem is Minimum Spanning Tree-- MST: building a tree that spans all the nodes in graph and the cost of edges on the tree is minimum possible. It is a polynomial problem. Prim and Kruskal each have well-known algorithms for the solution.
See http://en.wikipedia.org/wiki/Kruskal's_algorithm for Kruskal's algorithm.
Note: the problem is NP-complete when the tree is supposed to span a given proper subset of nodes instead of all nodes in the graph. This time it is known as the Steiner Minimal Tree problem.
Related
Suppose I am given an undirected and unweighted graph and a subset of nodes from the graph.
Now my target is to find the smallest tree or path that connects all the subset nodes. The order of nodes does not matter and neither the starting node. Any node can be the starting node.
My question is similar to Algorithm to find minimum spanning tree of chosen vertices but all the nodes have weight equal to 1. Hence, I am trying to find the tree with least number of links.
Extending the answer of the question you linked, you are now looking for an unweighted Steiner tree.
It is however also NP-hard, see this question on cstheory.stackexchange.
This question came to my mind when I see this question. For simplicity, we can limit our discussion to undirected, weighted, connected graphs. It is clear that Dijkstra cannot guarantee to produce a MST if we choose an arbitrary node from a graph as the source. However, is it guaranteed that there must exist one node in an undirected, weighted, connected graph, which will produce a MST for the graph if we choose it as the source and apply Dijkstra's algorithm? Maybe you can give a proof or a counterexample. Thanks!
However, is it guaranteed that there must exist one node in an
undirected, weighted, connected graph, which will produce a MST for
the graph if we choose it as the source and apply Dijkstra's
algorithm?
Nope, Dijkstra's algorithm minimizes the path weight from a single node to all other nodes. A minimum spanning tree minimizes the sum of the weights needed to connect all nodes together. There's no reason to expect that those disparate requirements will result in identical solutions.
Consider a complete graph where the sum of the weight of any two edges exceeds the weight of any single edge. That forces Dijkstra to always select the direct connection as the shortest path between two nodes. Then, if the lowest weight edges in the graph don't all originate from a single node, the minimum spanning tree won't be the same as any of the trees that Dijkstra will produce.
Here's an example:
The minimum spanning tree consists of the three edges with weight 3 (total weight 9). The trees returned by Dijkstra's algorithm will be whichever three edges connect directly to the source node (total weight 10 or 11).
write an algorithm to find a spanning tree that has the maximum number of leaves.
Write an algorithm to find a spanning tree with minimum number of nodes.
I am yet not able to come up with a solution for the following questions.
For the first part what I thought is to find the vertex with the highest degree and place it in the second last level such that the last level gets the maximum number of leaves.
Finding a spanning tree of a graph with maximum number of leaves is an NP-Complete problem. There is a reduction from the Dominating Set Problem which is NP-Complete.
Finding a spanning tree of a graph with minimum number of leaves is also an NP-Complete problem. Suppose if the graph has a Hamiltonian path then the graph has a spanning tree with just two leaves. Thus finding a spanning tree of a graph with minimum number of leaves is equivalent to finding whether a graph has a Hamiltonian path or not.
So for both the problems you need to develop approximation algorithms.
So my problem is the following:
I have an undirected (complete) weighted graph G=(V,E), and I would like to generate all the possible spanning trees with minimum number of leaves, i.e. with minimum number of vertices of degree 1. Let's call this kind of trees MIN_LEAF.
Possibly, I would like to directly generate, among all trees with minimum number of leaves, the one which has also the minimum total weight (please note that this is not necessarily a minimum spanning tree).
Is the problem of deciding if a tree T is a MIN_LEAF for a given graph G NP-complete?
If so, I wonder if some kind of heuristic algorithm exists (greedy or local search) which can at least give an approximate solution for this problem.
Thanks in advance.
The first problem you described - finding a spanning tree with the fewest number of leaves possible - is NP-hard. You can see this by reducing the Hamiltonian path problem to this problem: notice that a Hamiltonian path is a spanning tree of a graph and only has two leaf nodes, and that any spanning tree of a graph with exactly two leaf nodes must be a Hamiltonian path. That means that the NP-hard problem of determining whether a Hamiltonian path exists in a graph can be solved by finding the minimum-leaf spanning tree of the graph: the path exists if and only if the minimum-leaf spanning tree has exactly two leaves. The second problem you've described contains that first problem as a special case and therefore is going to also be NP-hard.
A quick Google search turned up the paper "On finding spanning trees with few leaves", which seems like it might be a good starting point for approximation algorithms (they have a 2-approximation for arbitrary graphs) and further reading on the subject.
I know a few algorithms that are able to find the lowest cost path for directed graph (just as Dijkstra and Floyd).
Is there any algorithm that works for non-directed graphs?
My problem is: I need to find the lowest cost path from a to b passing through all vertexes (undirected graph).
My problem is: I need to find the lowest cost path from a to b passing
through all vertexes (non-oriented graph)
This is the Traveling Salesman Problem, which is NP-Hard, so there is no known efficient solution to it.
However, if the graph is fairly small, there are some techniques to solve it optimally (in exponential time), like Dynamic Programming.
In general, changing an undirected graph to a directed one is fairly easy and is done by changing an undirected edge {u,v} to two directed edges (u,v) and (v,u)
Provided you have nonnegative edge values, you could consider every edge in an undirected graph as two edges in a directed graph, one pointed to and from connected vertices. Then you could use one of many algorithms including the ones you listed.