MinR Minimum Removal

Minimum Removal (MinR) is a technique used in graph theory to find the smallest number of edges that must be removed from a graph in order to transform it into an acyclic graph or tree. The problem of finding the minimum number of edges that must be removed from a graph to make it acyclic is known as the Minimum Feedback Arc Set (MFAS) problem.

The MFAS problem is NP-hard, meaning that it is computationally intractable to find an optimal solution for large graphs. However, there are a number of heuristic algorithms that can be used to find approximate solutions to the problem, including MinR.

In MinR, the graph is first represented as a directed graph, where each edge has a weight equal to its inverse frequency. The frequency of an edge is defined as the number of times it appears in all the cycles of the graph. The weight of an edge is defined as the reciprocal of its frequency.

The algorithm then calculates the shortest path between each pair of vertices in the graph using Dijkstra's algorithm. The shortest path is defined as the path with the lowest total weight, where the weight of a path is the sum of the weights of its edges.

Once the shortest paths have been calculated, the algorithm identifies the edges that belong to at least one cycle in the graph. These edges are assigned a score based on their frequency, which is equal to the sum of the weights of the shortest paths that include the edge.

The edges are then sorted in decreasing order of their score, and the algorithm iteratively removes the edge with the highest score until the graph becomes acyclic.

The MinR algorithm has a number of advantages over other heuristic algorithms for the MFAS problem. First, it is relatively simple to implement and can be applied to large graphs. Second, it has been shown to be highly effective in practice, producing solutions that are close to optimal for a wide range of graphs.

However, there are also some limitations to the MinR algorithm. One limitation is that it assumes that the graph is directed, which may not be the case for all types of graphs. Another limitation is that it assumes that the weights of the edges are independent of each other, which may not always be true.

Overall, MinR is a useful heuristic algorithm for solving the MFAS problem in practice, and can be used to efficiently identify the minimum number of edges that must be removed from a graph to make it acyclic. In addition to the basic MinR algorithm described above, there are also a number of variants and extensions that have been proposed in the literature. One extension is known as Incremental MinR, which is used to find the minimum number of edges that must be removed from a graph to make it acyclic after the addition of a new edge.

In Incremental MinR, the graph is first modified to include the new edge, and then the MinR algorithm is applied to the modified graph to find the minimum number of edges that must be removed to make it acyclic. The algorithm then removes the new edge and returns the minimum feedback arc set for the original graph.

Another variant of the MinR algorithm is known as the LP-MinR algorithm, which uses a linear programming approach to find an optimal solution to the MFAS problem. In LP-MinR, the feedback arc set is represented as a binary vector, and a linear programming formulation is used to find the feedback arc set that minimizes the total weight of the edges in the set.

LP-MinR has been shown to be highly effective in practice, producing solutions that are very close to optimal for a wide range of graphs. However, it is also more computationally intensive than the basic MinR algorithm, and may not be suitable for very large graphs.

Another variant of the MinR algorithm is known as the Local-MinR algorithm, which is designed to find a near-optimal solution to the MFAS problem in a decentralized network setting. In Local-MinR, each node in the network independently applies the MinR algorithm to its local neighborhood to find the edges that must be removed to make its local subgraph acyclic. The algorithm then merges the feedback arc sets from all the nodes to produce a global feedback arc set for the network.

Local-MinR has been shown to be effective in a number of real-world applications, such as wireless sensor networks and social networks. However, it also has some limitations, such as the need for a central coordinator to merge the feedback arc sets from the nodes.

In conclusion, the Minimum Removal (MinR) algorithm is a powerful heuristic algorithm for solving the Minimum Feedback Arc Set (MFAS) problem in graph theory. It is able to efficiently identify the minimum number of edges that must be removed from a graph to make it acyclic, and has been shown to be highly effective in practice for a wide range of graphs. While there are some limitations to the MinR algorithm, such as the assumption of a directed graph and the independence of edge weights, there are also a number of variants and extensions that have been proposed to address these limitations and make the algorithm even more effective.