I have programmed the minimum Hungarian algorithm for a bipartite graph, with Dijkstra's algorithm to find the minimum cost of a maximum matching. However, I want to use such an algorithm to implement the maximum Hungarian algorithm and don't know if it's correct to just negate the edges, because I don't know if the algorithm will handle it.
My implementation is based on the explanation on the following site: https://www.ics.uci.edu/~eppstein/163/lecture6b.pdf
Given G=(AUB, E), the idea is to label the vertices via an artificial start vertex s which has edges with unsaturated nodes in A, and run Dijkstra's algorithm from s in order to label each vertex, then after labeling each, the edges will be reweighted by their original weight minus the labels of the edge's endpoints.
I have read a lot of articles, and the only I could see is that a minimum Hungarian algorithm can be handled well with maximum cost by negating each edge, however, I am afraid that due to the fact that Dijkstra's algorithm doesn't handle negative edges well, it won't work.
First find the maximum weight in your graph. Then negate all of the weights and add the maximum weight to them. Adding the original maximum to all of the negated values makes them all positive.
You can also use INT_MAX (or whatever is equivalent to it in the programming language you're using) instead of the maximum weight. This skips the step of finding the maximum weight, but could make the first iteration of the Hungarian Algorithm take longer, or cause you to need an extra iteration of the algorithm to get the result. It probably doesn't make much of a difference either way and the performance difference will vary based on the particular weights in your graph.
Related
I am given a weighted graph and want to find a set of edges such that every node is only incident to one edge and such that the sum of the selected edge weights is maximized. As far as I know this problem is generally referred to as maximum weight matching and there exist fast approximations for it: https://web.eecs.umich.edu/~pettie/papers/ApproxMWM-JACM.pdf
However, for my application it would be better if only a certain ratio of nodes is paired. It's more important for my application that the nodes that get paired have a high weight between them. Leaving some nodes unpaired is no big problem.
Currently, I sort the weights between nodes in descending order and always select the edge with the highest weight until I have paired a certain number of nodes. Of course I ensure that pairs of nodes are mutually exclusive. This is only a 1/2 approximation to the original problem and it's probably even worse for the modified problem.
Could you please suggest an algorithm for this issue or tell me how this problem is called?
Some thoughts.
The greedy algorithm for this problem is a 2-approximation. Imagine that, during the execution of the greedy algorithm, we keep score versus an optimal solution in the following manner. Every time we add an edge to the greedy solution, we delete the incident edges in the optimal solution, which I claim must have weight no greater than the greedy edge. If no edge would be deleted from the optimal solution, we delete the two heaviest edges instead, which also I claim must have weight no greater than the greedy edge. Since the greedy solution must have at least half as many edges as the optimal solution, I claim that there are no optimal edges left at the end, and hence greedy is a 2-approximation because we never deleted more than 2x weight with each greedy edge.
A complete 20,000-vertex graph is right on that line where I don't know whether integer programming would be a good idea. I think it's still worth a try because it's easy enough.
There are polynomial-time algorithms for computing the maximum-weight independent set in the intersection of two matroids (for this problem, the matching matroid and the uniform matroid whose bases have size equal to the size of the desired matching). I don't know if they would be practical.
Given a graph that has negative weights but we know for sure it has no negative cycle:
Add a big enough constant to all weights so they are now positive and use Dijkstra's algorithm to find the smallest path.
Is the above correct if we don't have negative cycles? If we have negative cycles we can't use that algorithm since it will need to revisit the nodes Dijkstra marked as completed.
By adding a constant to all the weights you shall make paths with more number of edges more costly and paths with less number of edges relatively less costly hence disrupting the original problem.
So you can't apply Dijkstra even if there is no negative weight cycle.
I'm looking for an algorithm which I'm sure must have been studied, but I'm not familiar enough with graph theory to even know the right terms to search for.
In the abstract, I'm looking for an algorithm to determine the set of routes between reachable vertices [x1, x2, xn] and a certain starting vertex, when each edge has a weight and each route can only have a given maximum total weight of x.
In more practical terms, I have road network and for each road segment a length and maximum travel speed. I need to determine the area that can be reached within a certain time span from any starting point on the network. If I can find the furthest away points that are reachable within that time, then I will use a convex hull algorithm to determine the area (this approximates enough for my use case).
So my question, how do I find those end points? My first intuition was to use Dijkstra's algorithm and stop once I've 'consumed' a certain 'budget' of time, subtracting from that budget on each road segment; but I get stuck when the algorithm should backtrack but has used its budget. Is there a known name for this problem?
If I understood the problem correctly, your initial guess is right. Dijkstra's algorithm, or any other algorithm finding a shortest path from a vertex to all other vertices (like A*) will fit.
In the simplest case you can construct the graph, where weight of edges stands for minimum time required to pass this segment of road. If you have its length and maximum allowed speed, I assume you know it. Run the algorithm from the starting point, pick those vertices with the shortest path less than x. As simple as that.
If you want to optimize things, note that during the work of Dijkstra's algorithm, currently known shortest paths to the vertices are increasing monotonically with each iteration. Which is kind of expected when you deal with graphs with non-negative weights. Now, on each step you are picking an unused vertex with minimum current shortest path. If this path is greater than x, you may stop. There is no chance that you have any vertices with shortest path less than x from now on.
If you need to exactly determine points between vertices, that a vehicle can reach in a given time, it is just a small extension to the above algorithm. As a next step, consider all (u, v) edges, where u can be reached in time x, while v cannot. I.e. if we define shortest path to vertex w as t(w), we have t(u) <= x and t(v) > x. Now use some basic math to interpolate point between u and v with the coefficient (x - t(u)) / (t(v) - t(u)).
Using breadth first search from the starting node seems a good way to solve the problem in O(V+E) time complexity. Well that's what Dijkstra does, but it stops after finding the smallest path. In your case, however, you must continue collecting routes for your set of routes until no route can be extended keeping its weigth less than or equal the maximum total weight.
And I don't think there is any backtracking in Dijkstra's algorithm.
I have a directed graph that has all non-negative edges except the edge(s) that leave the source (S). There are no edges from any other vertices to the source. To find the shortest distance from source (S) to a vertex (T) in the graph, can I use Dijkstra's shortest path algorithm even though the edges leaving the source is negative?
Assuming only source-adjecent edges can have negative weights and there is no path back to the source from any of the source-adjecent nodes (as mentioned in the comment), you can just add a constant C onto all edges leaving the source to make them all non-negative. Then subtract C from the final result.
On a more general note, Dijkstra can be used to solve shortest-path in any graph with negative edge weights (but no negative cycles) after applying Johnson's reweighting algorithm (which is essentially Bellman-Ford, but needs to be performed only once).
Yes, you can use Dijkstra on that type of directed graph.
If you use already finished alghoritm for Dijsktra and it cannot use negative values, it can be good practise to find the lowest negative edge and add that number to all starting edges, therefore there is no-negative number at all. You substract that number after finishing.
If you code it yourself (which is acutally pretty easy and I recommend it to you), you almost does not change anything, just start with lowest value (as usual for Dijkstra) and allow it, that lowest value can be negative. It will work in your case.
The reason you generally can't use Dijkstra's algorithm for (directed) graphs with negative links is that Dijkstra's algorithm is greedy. It assumes that once you pick a vertex with minimum distance, there is no way it can later be reached by a smaller paths.
In your particular graph, after the very first step, you traverse all possible negative edges and Dijkstra's assumption actually holds from now on. Regardless of the fact that those vertices directly connected to start now have negative values, once you identify which has the minimum distance, it can never be reached again with a smaller distance (since all edges you would traverse from this point on would have a positive distance).
If you think about the conditions that dijkstra's algorithm puts upon the edges for the algorithm to work it is only that they are never decreasing after initialisation.
Thus, it actually doesn't matter if the first step is negative as from those several points onwards the function is constantly increasing and thus the correct output will be found (provided there is no way to get back to the start square.).
Given a weighted graph (directed or undirected) I need to find the cycle of the graph with the maximum weight.
The weight of a cycle being the sum of the weight of the edges of the graph.
It can be any cycle, not just base cycle for which we can
find all base cycle (see Algorithms to Identify All the Cycle Bases in a UnDirected Graph )
compute the weight of each base cycle and find the maximum
I could try to enumerate all cycles of the graph and then compute the maximum but the total number of cycles can be really big (if the graph is complete then any sequence of vertices where the first and last one are identical is a cycle).
Do you have any idea to find that maximum weight cycle without enumerating all cycles ?
If you need hypothesis on the graph (positives weights for example) please indicates them.
This is NP-Hard.
Hamiltonian Cycle problem can be reduced to this.
Given a graph for which we need to check if there exists a Hamiltonian Cycle or not, assign weight 1 to each edge.
Now run your algorithm to get the maximum weight cycle. If the weight is < n, then the original graph has no Hamiltonian cycle, otherwise it does.
If you can find the minimum weighted path in your specific case, just reverse the signs of all the weights and apply your algorithm. Of course you are making some unstated assumptions because Moron's argument is correct (no pun intended). The assumptions you are making could be positive weights or no negative weight cycles. I think you should make an effort to state them instead of letting people search in the infinite space of possible assumptions. As to hardness results, this is also hard to approximate in a number of way, check out this paper. The same paper contains several positive results for important types of graphs, but it's concerned with longest unweighted paths so my guess is that most algorithms in the paper won't directly help in your case. If you search for "Heavy cycles" you will find a number of interesting papers, but they are more mathematical in character. If your weights are small integers (up to a polynomial in the size of the graph), you can try and replace every edge with an unweighted path to reduce your problem to the unweighted case. I hope this helps to some degree, but you might have an open research problem on your hands.