Maximum flow in a network
In a graph where each edge has a given flow capacity the maximal flow between two vertices is calculated.
graph.maxflow(graph, source, target, capacity=NULL) graph.mincut(graph, source=NULL, target=NULL, capacity=NULL, value.only = TRUE)
- The input graph.
- The id of the source vertex.
- The id of the target vertex (sometimes also called sink).
- Vector giving the capacity of the edges. If this is
NULL(the default) then the
capacityedge attribute is used.
- Logical scalar, if
TRUEonly the minumum cut value is returned, if
FALSEthe edges in the cut and a the two (or more) partitions are also returned.
graph.maxflow calculates the maximum flow between two vertices
in a weighted (ie. valued) graph. A flow from
target is an assignment of non-negative real numbers to the
edges of the graph, satisfying two properties: (1) for each edge the
flow (ie. the assigned number) is not more than the capacity of the
capacity parameter or edge attribute), (2) for every
vertex, except the source and the target the incoming flow is the same
as the outgoing flow. The value of the flow is the incoming flow of
target vertex. The maximum flow is the flow of maximum
graph.mincut calculates the minimum st-cut between two vertices
in a graph (if the
target arguments are
given) or the minimum cut of the graph (if both
The minimum st-cut between
target is the
minimum total weight of edges needed to remove to eliminate all paths from
The minimum cut of a graph is the minimum total weight of the edges
needed to remove to separate the graph into (at least) two
components. (Which is to make the graph not strongly connected
in the directed case.)
The maximum flow between two vertices in a graph is the same as the minimum
calculate the same quantity, the only difference is that
graph.mincut can be invoked without giving the
target arguments and then minimum of all possible minimum
cuts is calculated.
For undirected graphs the Stoer-Wagner algorithm (see reference below) is used to calculate the minimum cut.
graph.maxflowa named list with components:
value A numeric scalar, the value of the maximum flow. flow A numeric vector, the flow itself, one entry for each edge. For undirected graphs this entry is bit trickier, since for these the flow direction is not predetermined by the edge direction. For these graphs the elements of the this vector can be negative, this means that the flow goes from the bigger vertex id to the smaller one. Positive values mean that the flow goes from the smaller vertex id to the bigger one. cut A numeric vector of edge ids, the minimum cut corresponding to the maximum flow. partition1 A numeric vector of vertex ids, the vertices in the first partition of the minimum cut corresponding to the maximum flow. partition2 A numeric vector of vertex ids, the vertices in the second partition of the minimum cut corresponding to the maximum flow.
graph.mincuta numeric constant, the value of the minimum cut, except if
value.only=FALSE. In this case a named list with components:
value Numeric scalar, the cut value. cut Numeric vector, the edges in the cut. partition1 The vertices in the first partition after the cut edges are removed. Note that these vertices might be actually in different components (after the cut edges are removed), as the graph may fall apart into more than two components. partition2 The vertices in the second partition after the cut edges are removed. Note that these vertices might be actually in different components (after the cut edges are removed), as the graph may fall apart into more than two components.
- Maximum flow
- Minimum cut
A. V. Goldberg and R. E. Tarjan: A New Approach to the Maximum Flow Problem Journal of the ACM 35:921-940, 1988. M. Stoer and F. Wagner: A simple min-cut algorithm, Journal of the ACM, 44 585-591, 1997.
E <- rbind( c(1,3,3), c(3,4,1), c(4,2,2), c(1,5,1), c(5,6,2), c(6,2,10)) colnames(E) <- c("from", "to", "capacity") g1 <- graph.data.frame(as.data.frame(E)) graph.maxflow(g1, source=V(g1)["1"], target=V(g1)["2"]) g <- graph.ring(100) graph.mincut(g, capacity=rep(1,vcount(g))) graph.mincut(g, value.only=FALSE, capacity=rep(1,vcount(g))) g2 <- graph( c(1,2,2,3,3,4, 1,6,6,5,5,4, 4,1) ) E(g2)$capacity <- c(3,1,2, 10,1,3, 2) graph.mincut(g2, value.only=FALSE)