Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

Source code for networkx.algorithms.flow.networksimplex

"""
Minimum cost flow algorithms on directed connected graphs.
"""

__all__ = ["network_simplex"]

from itertools import chain, islice, repeat
from math import ceil, sqrt
import networkx as nx
from networkx.utils import not_implemented_for


[docs]@not_implemented_for("undirected") def network_simplex(G, demand="demand", capacity="capacity", weight="weight"): r"""Find a minimum cost flow satisfying all demands in digraph G. This is a primal network simplex algorithm that uses the leaving arc rule to prevent cycling. G is a digraph with edge costs and capacities and in which nodes have demand, i.e., they want to send or receive some amount of flow. A negative demand means that the node wants to send flow, a positive demand means that the node want to receive flow. A flow on the digraph G satisfies all demand if the net flow into each node is equal to the demand of that node. Parameters ---------- G : NetworkX graph DiGraph on which a minimum cost flow satisfying all demands is to be found. demand : string Nodes of the graph G are expected to have an attribute demand that indicates how much flow a node wants to send (negative demand) or receive (positive demand). Note that the sum of the demands should be 0 otherwise the problem in not feasible. If this attribute is not present, a node is considered to have 0 demand. Default value: 'demand'. capacity : string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. weight : string Edges of the graph G are expected to have an attribute weight that indicates the cost incurred by sending one unit of flow on that edge. If not present, the weight is considered to be 0. Default value: 'weight'. Returns ------- flowCost : integer, float Cost of a minimum cost flow satisfying all demands. flowDict : dictionary Dictionary of dictionaries keyed by nodes such that flowDict[u][v] is the flow edge (u, v). Raises ------ NetworkXError This exception is raised if the input graph is not directed, not connected or is a multigraph. NetworkXUnfeasible This exception is raised in the following situations: * The sum of the demands is not zero. Then, there is no flow satisfying all demands. * There is no flow satisfying all demand. NetworkXUnbounded This exception is raised if the digraph G has a cycle of negative cost and infinite capacity. Then, the cost of a flow satisfying all demands is unbounded below. Notes ----- This algorithm is not guaranteed to work if edge weights or demands are floating point numbers (overflows and roundoff errors can cause problems). As a workaround you can use integer numbers by multiplying the relevant edge attributes by a convenient constant factor (eg 100). See also -------- cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost Examples -------- A simple example of a min cost flow problem. >>> G = nx.DiGraph() >>> G.add_node("a", demand=-5) >>> G.add_node("d", demand=5) >>> G.add_edge("a", "b", weight=3, capacity=4) >>> G.add_edge("a", "c", weight=6, capacity=10) >>> G.add_edge("b", "d", weight=1, capacity=9) >>> G.add_edge("c", "d", weight=2, capacity=5) >>> flowCost, flowDict = nx.network_simplex(G) >>> flowCost 24 >>> flowDict # doctest: +SKIP {'a': {'c': 1, 'b': 4}, 'c': {'d': 1}, 'b': {'d': 4}, 'd': {}} The mincost flow algorithm can also be used to solve shortest path problems. To find the shortest path between two nodes u and v, give all edges an infinite capacity, give node u a demand of -1 and node v a demand a 1. Then run the network simplex. The value of a min cost flow will be the distance between u and v and edges carrying positive flow will indicate the path. >>> G = nx.DiGraph() >>> G.add_weighted_edges_from( ... [ ... ("s", "u", 10), ... ("s", "x", 5), ... ("u", "v", 1), ... ("u", "x", 2), ... ("v", "y", 1), ... ("x", "u", 3), ... ("x", "v", 5), ... ("x", "y", 2), ... ("y", "s", 7), ... ("y", "v", 6), ... ] ... ) >>> G.add_node("s", demand=-1) >>> G.add_node("v", demand=1) >>> flowCost, flowDict = nx.network_simplex(G) >>> flowCost == nx.shortest_path_length(G, "s", "v", weight="weight") True >>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0]) [('s', 'x'), ('u', 'v'), ('x', 'u')] >>> nx.shortest_path(G, "s", "v", weight="weight") ['s', 'x', 'u', 'v'] It is possible to change the name of the attributes used for the algorithm. >>> G = nx.DiGraph() >>> G.add_node("p", spam=-4) >>> G.add_node("q", spam=2) >>> G.add_node("a", spam=-2) >>> G.add_node("d", spam=-1) >>> G.add_node("t", spam=2) >>> G.add_node("w", spam=3) >>> G.add_edge("p", "q", cost=7, vacancies=5) >>> G.add_edge("p", "a", cost=1, vacancies=4) >>> G.add_edge("q", "d", cost=2, vacancies=3) >>> G.add_edge("t", "q", cost=1, vacancies=2) >>> G.add_edge("a", "t", cost=2, vacancies=4) >>> G.add_edge("d", "w", cost=3, vacancies=4) >>> G.add_edge("t", "w", cost=4, vacancies=1) >>> flowCost, flowDict = nx.network_simplex( ... G, demand="spam", capacity="vacancies", weight="cost" ... ) >>> flowCost 37 >>> flowDict # doctest: +SKIP {'a': {'t': 4}, 'd': {'w': 2}, 'q': {'d': 1}, 'p': {'q': 2, 'a': 2}, 't': {'q': 1, 'w': 1}, 'w': {}} References ---------- .. [1] Z. Kiraly, P. Kovacs. Efficient implementation of minimum-cost flow algorithms. Acta Universitatis Sapientiae, Informatica 4(1):67--118. 2012. .. [2] R. Barr, F. Glover, D. Klingman. Enhancement of spanning tree labeling procedures for network optimization. INFOR 17(1):16--34. 1979. """ ########################################################################### # Problem essentials extraction and sanity check ########################################################################### if len(G) == 0: raise nx.NetworkXError("graph has no nodes") # Number all nodes and edges and hereafter reference them using ONLY their # numbers N = list(G) # nodes I = {u: i for i, u in enumerate(N)} # node indices D = [G.nodes[u].get(demand, 0) for u in N] # node demands inf = float("inf") for p, b in zip(N, D): if abs(b) == inf: raise nx.NetworkXError(f"node {p!r} has infinite demand") multigraph = G.is_multigraph() S = [] # edge sources T = [] # edge targets if multigraph: K = [] # edge keys E = {} # edge indices U = [] # edge capacities C = [] # edge weights if not multigraph: edges = G.edges(data=True) else: edges = G.edges(data=True, keys=True) edges = (e for e in edges if e[0] != e[1] and e[-1].get(capacity, inf) != 0) for i, e in enumerate(edges): S.append(I[e[0]]) T.append(I[e[1]]) if multigraph: K.append(e[2]) E[e[:-1]] = i U.append(e[-1].get(capacity, inf)) C.append(e[-1].get(weight, 0)) for e, c in zip(E, C): if abs(c) == inf: raise nx.NetworkXError(f"edge {e!r} has infinite weight") if not multigraph: edges = nx.selfloop_edges(G, data=True) else: edges = nx.selfloop_edges(G, data=True, keys=True) for e in edges: if abs(e[-1].get(weight, 0)) == inf: raise nx.NetworkXError(f"edge {e[:-1]!r} has infinite weight") ########################################################################### # Quick infeasibility detection ########################################################################### if sum(D) != 0: raise nx.NetworkXUnfeasible("total node demand is not zero") for e, u in zip(E, U): if u < 0: raise nx.NetworkXUnfeasible(f"edge {e!r} has negative capacity") if not multigraph: edges = nx.selfloop_edges(G, data=True) else: edges = nx.selfloop_edges(G, data=True, keys=True) for e in edges: if e[-1].get(capacity, inf) < 0: raise nx.NetworkXUnfeasible(f"edge {e[:-1]!r} has negative capacity") ########################################################################### # Initialization ########################################################################### # Add a dummy node -1 and connect all existing nodes to it with infinite- # capacity dummy edges. Node -1 will serve as the root of the # spanning tree of the network simplex method. The new edges will used to # trivially satisfy the node demands and create an initial strongly # feasible spanning tree. n = len(N) # number of nodes for p, d in enumerate(D): # Must be greater-than here. Zero-demand nodes must have # edges pointing towards the root to ensure strong # feasibility. if d > 0: S.append(-1) T.append(p) else: S.append(p) T.append(-1) faux_inf = ( 3 * max( chain( [sum(u for u in U if u < inf), sum(abs(c) for c in C)], (abs(d) for d in D), ) ) or 1 ) C.extend(repeat(faux_inf, n)) U.extend(repeat(faux_inf, n)) # Construct the initial spanning tree. e = len(E) # number of edges x = list(chain(repeat(0, e), (abs(d) for d in D))) # edge flows pi = [faux_inf if d <= 0 else -faux_inf for d in D] # node potentials parent = list(chain(repeat(-1, n), [None])) # parent nodes edge = list(range(e, e + n)) # edges to parents size = list(chain(repeat(1, n), [n + 1])) # subtree sizes next = list(chain(range(1, n), [-1, 0])) # next nodes in depth-first thread prev = list(range(-1, n)) # previous nodes in depth-first thread last = list(chain(range(n), [n - 1])) # last descendants in depth-first thread ########################################################################### # Pivot loop ########################################################################### def reduced_cost(i): """Returns the reduced cost of an edge i. """ c = C[i] - pi[S[i]] + pi[T[i]] return c if x[i] == 0 else -c def find_entering_edges(): """Yield entering edges until none can be found. """ if e == 0: return # Entering edges are found by combining Dantzig's rule and Bland's # rule. The edges are cyclically grouped into blocks of size B. Within # each block, Dantzig's rule is applied to find an entering edge. The # blocks to search is determined following Bland's rule. B = int(ceil(sqrt(e))) # pivot block size M = (e + B - 1) // B # number of blocks needed to cover all edges m = 0 # number of consecutive blocks without eligible # entering edges f = 0 # first edge in block while m < M: # Determine the next block of edges. l = f + B if l <= e: edges = range(f, l) else: l -= e edges = chain(range(f, e), range(l)) f = l # Find the first edge with the lowest reduced cost. i = min(edges, key=reduced_cost) c = reduced_cost(i) if c >= 0: # No entering edge found in the current block. m += 1 else: # Entering edge found. if x[i] == 0: p = S[i] q = T[i] else: p = T[i] q = S[i] yield i, p, q m = 0 # All edges have nonnegative reduced costs. The current flow is # optimal. def find_apex(p, q): """Find the lowest common ancestor of nodes p and q in the spanning tree. """ size_p = size[p] size_q = size[q] while True: while size_p < size_q: p = parent[p] size_p = size[p] while size_p > size_q: q = parent[q] size_q = size[q] if size_p == size_q: if p != q: p = parent[p] size_p = size[p] q = parent[q] size_q = size[q] else: return p def trace_path(p, w): """Returns the nodes and edges on the path from node p to its ancestor w. """ Wn = [p] We = [] while p != w: We.append(edge[p]) p = parent[p] Wn.append(p) return Wn, We def find_cycle(i, p, q): """Returns the nodes and edges on the cycle containing edge i == (p, q) when the latter is added to the spanning tree. The cycle is oriented in the direction from p to q. """ w = find_apex(p, q) Wn, We = trace_path(p, w) Wn.reverse() We.reverse() if We != [i]: We.append(i) WnR, WeR = trace_path(q, w) del WnR[-1] Wn += WnR We += WeR return Wn, We def residual_capacity(i, p): """Returns the residual capacity of an edge i in the direction away from its endpoint p. """ return U[i] - x[i] if S[i] == p else x[i] def find_leaving_edge(Wn, We): """Returns the leaving edge in a cycle represented by Wn and We. """ j, s = min( zip(reversed(We), reversed(Wn)), key=lambda i_p: residual_capacity(*i_p) ) t = T[j] if S[j] == s else S[j] return j, s, t def augment_flow(Wn, We, f): """Augment f units of flow along a cycle represented by Wn and We. """ for i, p in zip(We, Wn): if S[i] == p: x[i] += f else: x[i] -= f def trace_subtree(p): """Yield the nodes in the subtree rooted at a node p. """ yield p l = last[p] while p != l: p = next[p] yield p def remove_edge(s, t): """Remove an edge (s, t) where parent[t] == s from the spanning tree. """ size_t = size[t] prev_t = prev[t] last_t = last[t] next_last_t = next[last_t] # Remove (s, t). parent[t] = None edge[t] = None # Remove the subtree rooted at t from the depth-first thread. next[prev_t] = next_last_t prev[next_last_t] = prev_t next[last_t] = t prev[t] = last_t # Update the subtree sizes and last descendants of the (old) acenstors # of t. while s is not None: size[s] -= size_t if last[s] == last_t: last[s] = prev_t s = parent[s] def make_root(q): """Make a node q the root of its containing subtree. """ ancestors = [] while q is not None: ancestors.append(q) q = parent[q] ancestors.reverse() for p, q in zip(ancestors, islice(ancestors, 1, None)): size_p = size[p] last_p = last[p] prev_q = prev[q] last_q = last[q] next_last_q = next[last_q] # Make p a child of q. parent[p] = q parent[q] = None edge[p] = edge[q] edge[q] = None size[p] = size_p - size[q] size[q] = size_p # Remove the subtree rooted at q from the depth-first thread. next[prev_q] = next_last_q prev[next_last_q] = prev_q next[last_q] = q prev[q] = last_q if last_p == last_q: last[p] = prev_q last_p = prev_q # Add the remaining parts of the subtree rooted at p as a subtree # of q in the depth-first thread. prev[p] = last_q next[last_q] = p next[last_p] = q prev[q] = last_p last[q] = last_p def add_edge(i, p, q): """Add an edge (p, q) to the spanning tree where q is the root of a subtree. """ last_p = last[p] next_last_p = next[last_p] size_q = size[q] last_q = last[q] # Make q a child of p. parent[q] = p edge[q] = i # Insert the subtree rooted at q into the depth-first thread. next[last_p] = q prev[q] = last_p prev[next_last_p] = last_q next[last_q] = next_last_p # Update the subtree sizes and last descendants of the (new) ancestors # of q. while p is not None: size[p] += size_q if last[p] == last_p: last[p] = last_q p = parent[p] def update_potentials(i, p, q): """Update the potentials of the nodes in the subtree rooted at a node q connected to its parent p by an edge i. """ if q == T[i]: d = pi[p] - C[i] - pi[q] else: d = pi[p] + C[i] - pi[q] for q in trace_subtree(q): pi[q] += d # Pivot loop for i, p, q in find_entering_edges(): Wn, We = find_cycle(i, p, q) j, s, t = find_leaving_edge(Wn, We) augment_flow(Wn, We, residual_capacity(j, s)) # Do nothing more if the entering edge is the same as the leaving edge. if i != j: if parent[t] != s: # Ensure that s is the parent of t. s, t = t, s if We.index(i) > We.index(j): # Ensure that q is in the subtree rooted at t. p, q = q, p remove_edge(s, t) make_root(q) add_edge(i, p, q) update_potentials(i, p, q) ########################################################################### # Infeasibility and unboundedness detection ########################################################################### if any(x[i] != 0 for i in range(-n, 0)): raise nx.NetworkXUnfeasible("no flow satisfies all node demands") if any(x[i] * 2 >= faux_inf for i in range(e)) or any( e[-1].get(capacity, inf) == inf and e[-1].get(weight, 0) < 0 for e in nx.selfloop_edges(G, data=True) ): raise nx.NetworkXUnbounded("negative cycle with infinite capacity found") ########################################################################### # Flow cost calculation and flow dict construction ########################################################################### del x[e:] flow_cost = sum(c * x for c, x in zip(C, x)) flow_dict = {n: {} for n in N} def add_entry(e): """Add a flow dict entry. """ d = flow_dict[e[0]] for k in e[1:-2]: try: d = d[k] except KeyError: t = {} d[k] = t d = t d[e[-2]] = e[-1] S = (N[s] for s in S) # Use original nodes. T = (N[t] for t in T) # Use original nodes. if not multigraph: for e in zip(S, T, x): add_entry(e) edges = G.edges(data=True) else: for e in zip(S, T, K, x): add_entry(e) edges = G.edges(data=True, keys=True) for e in edges: if e[0] != e[1]: if e[-1].get(capacity, inf) == 0: add_entry(e[:-1] + (0,)) else: c = e[-1].get(weight, 0) if c >= 0: add_entry(e[:-1] + (0,)) else: u = e[-1][capacity] flow_cost += c * u add_entry(e[:-1] + (u,)) return flow_cost, flow_dict