LightGraphsFlows

Flow algorithms on top of LightGraphs.jl, including maximum_flow, multiroute_flow and mincost_flow. See Maximum flow problem for a detailed description of the problem.

Documentation for this package is available here. For an overview of JuliaGraphs, see this page.

Usage

Maxflow

julia> flow_graph = lg.DiGraph(8) # Create a flow-graph
julia> flow_edges = [
(1,2,10),(1,3,5),(1,4,15),(2,3,4),(2,5,9),
(2,6,15),(3,4,4),(3,6,8),(4,7,16),(5,6,15),
(5,8,10),(6,7,15),(6,8,10),(7,3,6),(7,8,10)
]

julia> capacity_matrix = zeros(Int, 8, 8)  # Create a capacity matrix

julia> for e in flow_edges
    u, v, f = e
    lg.add_edge!(flow_graph, u, v)
    capacity_matrix[u,v] = f
end

julia> f, F = maximum_flow(flow_graph, 1, 8) # Run default maximum_flow (push-relabel) without the capacity_matrix

julia> f, F = maximum_flow(flow_graph, 1, 8, capacity_matrix) # Run default maximum_flow with the capacity_matrix

julia> f, F = maximum_flow(flow_graph, 1, 8, capacity_matrix, algorithm=EdmondsKarpAlgorithm()) # Run Edmonds-Karp algorithm

julia> f, F = maximum_flow(flow_graph, 1, 8, capacity_matrix, algorithm=DinicAlgorithm()) # Run Dinic's algorithm

julia> f, F, labels = maximum_flow(flow_graph, 1, 8, capacity_matrix, algorithm=BoykovKolmogorovAlgorithm()) # Run Boykov-Kolmogorov algorithm

Multi-route flow

julia> flow_graph = lg.DiGraph(8) # Create a flow graph

julia> flow_edges = [
(1, 2, 10), (1, 3, 5),  (1, 4, 15), (2, 3, 4),  (2, 5, 9),
(2, 6, 15), (3, 4, 4),  (3, 6, 8),  (4, 7, 16), (5, 6, 15),
(5, 8, 10), (6, 7, 15), (6, 8, 10), (7, 3, 6),  (7, 8, 10)
]

julia> capacity_matrix = zeros(Int, 8, 8) # Create a capacity matrix

julia> for e in flow_edges
    u, v, f = e
    add_edge!(flow_graph, u, v)
    capacity_matrix[u, v] = f
end

julia> f, F = multiroute_flow(flow_graph, 1, 8, capacity_matrix, routes = 2) # Run default multiroute_flow with an integer number of routes = 2

julia> f, F = multiroute_flow(flow_graph, 1, 8, capacity_matrix, routes = 1.5) # Run default multiroute_flow with a noninteger number of routes = 1.5

julia> points = multiroute_flow(flow_graph, 1, 8, capacity_matrix) # Run default multiroute_flow for all the breaking points values

julia> f, F = multiroute_flow(points, 1.5) # Then run multiroute flow algorithm for any positive number of routes

julia> f = multiroute_flow(points, 1.5, valueonly = true)

julia> f, F, labels = multiroute_flow(flow_graph, 1, 8, capacity_matrix, algorithm = BoykovKolmogorovAlgorithm(), routes = 2) # Run multiroute flow algorithm using Boykov-Kolmogorov algorithm as maximum_flow routine

Mincost flow

Mincost flow is solving a linear optimization problem and thus requires a LP solver defined by MathProgBase.jl.

using Clp: ClpSolver
g = lg.DiGraph(6)
lg.add_edge!(g, 5, 1)
lg.add_edge!(g, 5, 2)
lg.add_edge!(g, 3, 6)
lg.add_edge!(g, 4, 6)
lg.add_edge!(g, 1, 3)
lg.add_edge!(g, 1, 4)
lg.add_edge!(g, 2, 3)
lg.add_edge!(g, 2, 4)
w = zeros(6,6)
w[1,3] = 10.
w[1,4] = 5.
w[2,3] = 2.
w[2,4] = 2.
# v2 -> sink have demand of one
demand = spzeros(6,6)
demand[3,6] = 1
demand[4,6] = 1
capacity = ones(6,6)

flow = mincost_flow(g, capacity, demand, w, ClpSolver(), 5, 6)