Chapter 6 Maximum Flow Problems

1
Chapter 6Maximum Flow Problems

• Flows and Cuts
• Augmenting Path Algorithm

2
Definitions/Assumptions

• Each arc (i, j) has an arc capacity uij.
• The problem is to find the maximum flow from the
  source node, s, to the sink node, t.
• Assumptions
• The network is directed
• All capacities are nonnegative integers
• There is no directed path from s to t composed
  only of infinite capacity arcs
• Each arc (i, j) has a corresponding reverse arc
  (j, i) (could have 0 capacity)
• There are no parallel arcs

3
Residual Network (see pp. 44-46)

• Given a set of flows x xij, the residual
  network tells what changes are feasible. For
  each arc (i, j), G(x) contains
• A forward arc with residual capacity rij uij
  xij (if gt 0)
• A backward arc with capacity rij xij (if gt 0)
• The residual capacity of an s-t cut is the sum of
  the residual capacities of forward arcs in the
  cut.
• Facts
• The value of any flow is less than or equal to
  the capacity of any s-t cut in the network.
• The additional flow that can be sent from s to t
  is no greater than the residual capacity of any
  s-t cut.

4
Generic Augmenting Path Algorithm

• algorithm augmenting path
• begin
• x 0
• while G(x) contains a directed path from node s
  to node t do
• begin
• Identify an augmenting path P from node s to
  node t
• augment ? units of flow along P and update
  G(x)
• end
• end

5
Augmenting Paths

• An augmenting path is a directed path from the
  source to the sink in the residual network
• Its capacity is the minimum (residual) capacity
  of the arcs it contains
• If the network contains an augmenting path, then
  we can send more units of flow from the source to
  the sink
• An increase of ? units on arc (i, j) in the
  residual network corresponds to, in the original
  network, increasing xij by ? units, or decreasing
  xji by ? units, or increasing xij by a? units
  while decreasing xji by (1-a)? units, 0 lt a lt 1
  (i.e., an aug. path may actually subtract flows
  from some arcs)
• Residual capacities do not uniquely determine the
  flows in the original network. We will use the
  convention
• if uij ? rij, set xij uij - rij and xji 0
• otherwise, set xij 0 and xji rij - uij

6
Labeling Algorithm

• How to find an augmenting path?
• Use the search algorithm (from Chapter 3) to find
  nodes reachable from the source node in the
  residual network. If the sink is reachable then
  a path from the source can be traced backward
  using predecessor indices.
• If the search procedure fails to label the sink
  node as reachable, then it will have identified
  an s-t cut in the residual network with rij 0
  for each arc (i, j) in the cut. Therefore, xij
  uij for each arc in the cut, and the total flow
  from s to t equals the capacity of this cut.

7
The Labeling Algorithm (main)
8
Procedure augment
9
Max-Flow Min-Cut Theorem

• Since the labeling algorithm stops when it
  identifies an s-t cut whose capacity equals the
  flow from s to t,
• and since the capacity of any s-t cut is an upper
  bound on the total value of any feasible flow, it
  follows that
• (Max-Flow Min-Cut Theorem) The maximum value of
  the flow from a source node s to a sink node t in
  a capacitated network equals the minimum capacity
  among all s-t cuts.
• Also,
• A flow is a maximum flow if and only if the
  corresponding residual network contains no
  augmenting path, and
• If all arc capacities are integer, the maximum
  flow problem has an integer maximum flow.

10
Ch. 7 Distance Labels

• In maximum flow problems, measure the distance of
  a node from the sink node as the smallest number
  of arcs on any directed path from this node to
  the sink.
• t 4 d(1) 3, d(2) 1, d(3) 2, d(4) 0
• The shortest augmenting path algorithm identifies
  the augmenting path in the residual network
  containing the fewest arcs and runs in O(n2m)
  time (labeling algorithm is O(nmU), where U is
  the largest finite arc capacity).

11
Preflow-Push Algorithm

• Augmenting path algorithms can take excessive
  time by repeatedly sending flows down the same
  subpaths (see Fig.7.10). Sending a flow along a
  path takes O(n) time in the worst case.
• Note that augmenting path algorithms always
  maintain a feasible flow (satisfies capacity and
  flow conservation constraints).
• The preflow-push algorithm always obeys capacity
  constraints but does not satisfy the conservation
  constraints until it reaches the optimal
  solution.
• A preflow is a set of arc flows in which the
  total flow into a node may exceed the total flow
  out of the node
• A push is the simple operation of sending flow
  down a single arc.

12
Excesses

• Given a flow x, the excess of each node i is
  given by
• In a preflow, e(i) ? 0 for each i ? s. Node s is
  the only node with a negative excess (because no
  arcs are directed into it). Nodes i ? t with
  e(i) gt 0 are called active.
• Active nodes indicate infeasibility the
  algorithm moves toward feasibility by trying to
  push the excess flow to the active nodes
  neighbors in particular, to nodes closer to the
  sink (measured by distance).
• An admissible arc is an arc (i, j) for which d(i)
  d(j) 1.

13
Preflow-Push Subroutines

• procedure preprocess
• begin
• x 0
• compute the exact distance labels d(i)
• xsj usj for each arc (s, j) ? A(s)
• d(s) n
• end
• procedure push/relabel(i)
• begin
• if the network contains an admissible arc (i, j)
  then
• push ? mine(i), rij units of flow from i to
  j
• else replace d(i) by mind(j) 1 (i, j) ?
  A(i) and rij gt 0
• end

14
Preflow-Push Algorithm

• algorithm preflow-push
• begin
• preprocess
• while the network contains an active node do
• begin
• select an active node i
• push/relabel(i)
• end
• end

15
Termination Complexity

• The preflow-push algorithm stops when all the
  excess is collected at the source or the sink,
  i.e., the current preflow is a feasible flow.
  The excess at the sink is the maximum flow value.
• This generic version runs in O(n2m) time.
  Depending on how the list of active nodes is
  maintained, variations of it run in O(n3),
  O(n2m1/2), or O(nm n2 log U) time, which may
  be smaller depending on the instance.