I am trying to solve this problem : Jobs.
So far i have thought that the problem is same as the Assignment Problem with the distributors and districts represented as a bipartite graph and the edges representing the probability. But here we would need to maximize the product rather than the sum of weights of matched edges.
One idea that came to my mind was to change each edge weight to log ( weight ). Then the problem essentially changes to finding the maximum sum, which is can then be solved using the algorithms for Assignment Problem. But this poses a problem, since applying log will make the edge weights non-integer, something which i think the Hungarian Algorithm does not work for.
Please suggest some other alternative approach.
In theory, the Hungarian algorithm works fine with real weights.
In practice, it's possible that, since most integer logarithms cannot be represented exactly as floating-point numbers, it could come to pass that rounding would change the optimal solution. There are ways to deal with that even so, but it's unlikely that you'll need them for this programming contest problem.
Related
is there an algorithm for finding all the independent sets of an directed graph ?
From what i've read an independent set represents a set formed by the nodes that are not adjacent.
So for this example I would have {1} {2} {1,3}
So how is possible to find all of them, I am thinking about something recursive but I don't really know the algorithm, if someone could point me in the right direction it would be much appreciated !
Thank you!
Typical way to find independent sets is to consider the complement of a graph. A complement of a graph is defined as a graph with the same set of vertices and an edge between a pair if and only if there is no edge between them in the original graph. An independent set in the graph corresponds to a clique in the complements. Finding all the cliques is exponential in complexity so you can not improve brute force much. Still I believe considering the complement of the graph may make the problem easier to deal with.
Other than complement and finding cliques, I can also think about "Graph Coloring", you color the vertices somehow that no two adjacent vertices have the same color (you can do it with a very simple heuristic algorithm like SL = Smallest Last), and then choose vertices in every color as a subset (as a maximal independent subset).
The only problem is that there are probably too many ways of coloring a graph. You have to keep all the found (maximal) independent sets and move on until you get enough sets!
The Bron–Kerbosch algorithm is commonly used for this problem, see the Wikipedia article for a description and pseudocode that can be turned into a useable program without too much problem. The size of output is, in the worst case, exponential in the number of vertices, but brute force will always be exponential while BK will be polynomial if the output is polynomial. In other words if you know that the output will be reasonable then BK will produce it in a reasonable time. This is an active area of research and there are a number of other algorithms that do the same thing with varying efficiency depending of the type and size of graph. There are applications in several areas, in particular genetics.
I'm aware of there's a lot of similar topics. But most of them left me some doubts in my case. What I want to do is find perfect matching (or as close to perfect as possible in case there's no perfect matching of course) and then from all of those matchings where you are able to match k out of n vertexes (where k is highest possible), I want to choose the highest possible total weight.
So simply put what I'm saying is following priority:
Match as many vertexes as possible
Because (non weighted) maximum matching in most cases is unambiguous, I want choose the one that have the biggest sum of weights on edges. If there are several of them with same weight it doesn't matter which would be chosen.
I've heard about Ford-Fulkerson algorithm. Is it working in the way I describe it or I need other algorithm?
If you're implementing this yourself, you probably want to use the Hungarian algorithm. Faster algorithms exist but aren't as easy to understand or implement.
Ford-Fulkerson is a maximum flow algorithm; you can use it easily to solve unweighted matching. Turning it into a weighted matcing algorithm requires an additional trick; with that trick, you wind up with the Hungarian algorithm.
You can also use a min-cost flow algorithm to do weighted bipartite matching, but it might not work quite as well. There's also the network simplex method, but it seems to be mostly of historical interest.
My Problem
Whether there's an efficient algorithm to find a max-weight (or min-weight) k-clique in a complete k-partite graph (a graph in which vertices are adjacent if and only if they belong to different partite sets according to wikipedia)?
More Details about the Terms
Max-weight Clique: Every edge in the graph has a weight. The weight of a clique is the sum of the weights of all edges in the clique. The goal is to find a clique with the maximum weight.
Note that the size of the clique is k which is the largest possible clique size in a complete k-partite graph.
What I have tried
I met this problem during a project. Since I am not a CS person, I am not sure about the complexity etc.
I have googled several related papers but none of them deals with the same problem. I have also programmed a greedy algorithm + simulated annealing to deal with it (the result seems not good). I have also tried something like Dynamic Programming (but it does not seem efficient). So I wonder whether the exact optimal can be computed efficiently. Thanks in advance.
EDIT Since my input can be really large (e.g. the number of vertices in each clique is 2^k), I hope to find a really fast algorithm (e.g. polynomial of k in time) that works out the optimal result. If it's not possible, can we prove some lower bound of the complexity?
Generalized Maximum Clique Problem (GMCP)
I understand that you are looking for the Generalized Maximum/ minimum Clique Problem (GMCP), where finding the clique with maximum score or minimum cost is the optimization problem.
This problem is a NP-Hard problem as shown in Generalized network design problems, so there is currently no polynomial time exact solution to your problem.
Since, there is no known polynomial solution to your problem, you have 2 choices. Reducing the problem size to find the exact solution or to find an estimated solution by relaxing your problem and it leads you to a an estimation to the optimal solution.
Example and solution for the small problem size
In small k-partite graphs (in our case k is 30 and each partite has 92 nodes), we were able to get the optimal solution in a reasonable time by a heavy branch and bounding algorithm. We have converted the problem into another NP-hard problem (Mixed Integer Programming), reduced number of integer variables, and used IBM Cplex optimizer to find the optimal solution to GMCP.
You can find our project page and paper useful. I can also share the code with you.
How to estimate the solution
One straight forward estimation to this NP-Hard problem is relaxing the Mixed Integer Programming problem and solve it as a linear programming problem. Of course it will give you an estimation of the solution, but still you might get a reasonable answer in practice.
More general problem (Generalized Maximum Multi Clique Problem)
In another work, we solve the Generalized Maximum Multi Clique Problem (GMMCP), where maximizing the score or minimizing the cost of selecting multiple k-cliques in a complete k-partite graph is in interest. You can find the project page by searching for GMMCP Tracking.
The maximum clique problem in a weighted graph in general is intractable. In your case, if the graph contains N nodes, you can enumerate through all possible k-cliques in N ** k time. If k is fixed (don't know if it is), your problem is trivially polynomially solvable, as this is a polynomial in N. I don't believe the problem to be tractable if k is a free parameter because I can't see how the assumption of a k-partite graph would make the problem significantly simpler from the general one.
How hard your problem is in practice depends also on how the weights are distributed. If all the weights are very near to each others, i.e. the difference between "best" and "good" is relatively small, the problem is very hard. If you have wildly different weights on the edges, the problem can be easier, because a greedy algorithm can give you a good "initial" solution, and you can use that and subsequent good solutions to limit your combinatorial search using the well-known branch-and-bound method.
I got two sets of points S and V, both have the size n. I want to link the two sets so that every point in S links to one and only one point in V. The cost to link two points is defined as the Euclidean distance between the two points. There should be n! possible ways to link. So how to find the way of minimum cost? (in an efficient way)
This is an assignment problem. You can solve it with the Hungarian Method. There are implementations of this in python. You can also solve the problem with any linear programming solver. The LP formulation will always give you an integer solution.
I have a problem which I like and I love to think about solutions, but I'm stuck unfortunately. I hope you like it too. The problem states:
I have two lists of 2D points(say A and B) and need to pair up points from A with points from B, under the condition that the sum of the distances in all pairs is minimal. A pair contains one point from A and one from B, a point can be used only once, and as many as possible pairs should be created(i.e. min(length(A), length(B))).
I've made a simple example, where color denotes which list the point is from, and the black connections are the solution.
Although this is a nice problem and I suspect is NP-hard, it gets nicer. I can build on existing solutions. Suppose I have two lists and the corresponding solution(i.e. the set of pairs), then the problem I need to solve is to reoptimalize that solution when a point is added to or removed from either list.
I've unfortunately not been able to come up with any non-brute force algorithm yielding the optimal solution. I hope you can. Any algorithm is appreciated in any (pseudo) language, preferably C#.
This problem is solvable in polynomial time via the Hungarian algorithm. To get a square matrix, add dummy entries to the shorter list at "distance 0" from everything.
Your problem is an instance of the weighted minimum maximal matching problem (as described in this Wikipedia article). There is no polynomial-time algorithm even for the unweighted problem (all distances equal). There are efficient algorithms to approximately solve it in polynomial time (within a factor of 2).
This is the minimum weight Euclidean bipartite matching problem. There is a O(n^(2+epsilon)) algorithm.