Computing Max Flows Through Cut Nodes

International Journal of Computer Applications, 2013

In this paper a new approximation algorithm for calculating the min-cut tree of an undirected edgeweighted graph has been proposed. This algorithm runs in 2 2 (.log .) O V V V d  , where V is the number of vertices in the given graph and d is the degree of the graph. It is a significant improvement over time complexities of existing solutions. However, because of an assumption it does not produce correct result for all sort of graphs but for the dense graphs success rate is more than 90%. Moreover in the unsuccessful cases, the deviation from actual result is very less and for most of the pairs we obtain correct values of max-flow or mincut. This algorithm is implemented in JAVA language and checked for many input cases.

2000

This work presents an algorithm for computing the maximum flow and minimum cut of undirected graphs, based on the well-known algorithm presented by Ford and Fulkerson for directed graphs. The new algorithm is equivalent to just applying Ford and Fulkerson algorithm to the directed graph obtained from original graph but with two directed arcs for each edge in the graph,

2018

The generalized node-capacitated maximum flow problem is to send the maximum amount of flow from a source node to a sink node in a directed network with node capacities, where flow on a given arc exerts a workload proportional to its amount on its tail and head nodes; and sum of all such workloads on each node cannot exceed the node’s capacity. The problem has important applications and efficient solution methods are essential. We develop an efficient Lagrangian relaxation and branch and bound based method that solves the problem as a series of shortest path problems.

Discrete Optimization, 2008

The Maximum Flow Problem with flow width constraints is an NP-hard problem. Two models are proposed: the first model is a compact node-arc model using two flow conservation blocks per path. For each path, one block defines the path while the other one sends the right amount of flow on it. The second model is an extended arc-path model, obtained from the first model after a Dantzig-Wolfe reformulation. It is an extended model as it relies on the set of all the paths between the source and the sink nodes. Some symmetry breaking constraints are used to improve the model. A Branch and Price algorithm is proposed to solve the problem. The column generation procedure reduces to the computation of a shortest path whose cost depends on weights on the arcs and on the path capacity. A polynomial-time algorithm is proposed to solve this subproblem. Computational results are shown on a set of medium-sized instances to show the effectiveness of our approach.

Networks, 2006

We are given a directed network G = (V, A, u) with vertex set V , arc set A, a source vertex s ∈ V , a destination vertex t ∈ V , a finite capacity vector u = {u ij } ij∈A , and a positive integer m ∈ Z +. The multiroute maximum flow problem (m-MFP) generalizes the ordinary maximum flow problem by seeking a maximum flow from s to t subject to not only the regular flow conservation constraints at the vertices (except s and t) and the flow capacity constraints at the arcs, but also the extra constraints that any flow must be routed along m arc-disjoint s-t paths. In this paper, we devise two new combinatorial algorithms for m-MFP. One is based on Newton's method and another is based on augmenting-path technique. We also show how the Newton-based algorithm unifies two existing algorithms, and how the augmenting-path algorithm is strongly polynomial for case m = 2.

Computers & Operations Research, 2012

The capacity scaling algorithm for the maximum flow problem runs in Oðnm log UÞ time where n is the number of nodes, m is the number of arcs, and U is the largest arc capacity in the network. The twophase capacity scaling algorithm reduces this bound to Oðnm logðU=nÞÞ. This running time is achieved with the restriction that flows are pushed on individual arcs while paths are being identified, but this causes slower empirical run times compared to the single-phase capacity scaling algorithm. In this research, we prove that the two-phase algorithm runs in the same theoretical time without the mentioned restriction. We also show that in practice, it runs significantly faster than the single-phase capacity scaling algorithm.

SIAM Journal on Computing, 1989

The classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter, and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition. Finding the desired parameter value requires solving a sequence of related maximum flow problems. In this paper it is shown that the recent maximum flow algorithm of Goldberg and Tarjan can be extended to solve an important class of such parametric maximum flow problems, at the cost of only a constant factor in its worst-case time bound. Faster algorithms for a variety of combinatorial optimization problems follow from the result.

Open Computer Science, 2016

An upgraded version of Ford-Fulkerson algorithm is proposed, which allows to determine the maximum flow and the distribution flow in branches of the network. Results of simulation using this algorithm are given. They show its effectiveness in comparison with the other variants of flow distribution in the network at change of the capacity of branches randomly in time.

SIAM Journal on Computing, 1979

Efficient algorithms for finding maximum flow in planar networks are presented. These algorithms take advantage of the planarity and are superior to the most efficient algorithms to date, If the source and the terminal are on the same face, an algorithm of Berge is improved and its time complexity is reduced to O(n log n). In the general case, for a given D > 0 a flow of value D is found if one exists; otherwise, it is indicated thatno such flow exists. This algorithm requires O(n 2 log n) time. If the network is undirected a minimum cut may be found in O(n log n) time. All algorithms require O(n)space.